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The Quantitative Reasoning domain tests your ability to use numbers and mathematical concepts to solve mathematical problems, as well as your ability to analyze data presented in a variety of ways, such as in table or graph form. Only a basic knowledge of mathematics is required (the material studied up to 9th or 10th grades in most Israeli high schools).
All of the Quantitative Reasoning problems take the form of multiple-choice questions, that is, a question followed by four possible responses, only one of which is the correct answer.
The Quantitative Reasoning sections consist of two categories of questions – Questions and Problems, and Graph or Table Comprehension.
Questions and Problems cover a variety of subjects taken from algebra and geometry. Some of the questions are presented in mathematical terms; others are word problems, which you must translate into mathematical terms before solving.
Graph or Table Comprehension questions relate to information appearing in a graph or a table. A graph presents data in graphic form, such as a bar chart, line graph or scatter plot. A table presents data arranged in columns and rows.
In general, all questions of a given type are arranged in ascending order of difficulty. The easier questions, requiring relatively less time to solve, appear first, with the questions becoming progressively more difficult and requiring more time to solve.
The figures accompanying some of the questions are not necessarily drawn to scale. Do not rely solely on the figure's appearance to deduce line length, angle size, and so forth. However, if a line in a figure appears to be straight, you may assume that it is, in fact, a straight line.
A Formula Page appears at the beginning of each Quantitative Reasoning section. This page contains instructions, general comments and mathematical formulas, which you may refer to during the test. The Formula Page also appears in the Guide (on the next page) and in the Quantitative Reasoning sections of the practice test. You should familiarize yourself with its contents prior to taking the test.
Pages 38-66 contain a review of basic mathematical concepts, covering much of the material upon which the questions in the Quantitative Reasoning sections are based. The actual test may, however, include some questions involving mathematical concepts and theorems that do not appear on these pages.
Pages 67-82 contain examples of different types of questions, each followed by the answer and a detailed explanation.
QUANTITATIVE REASONING
Guide For Examinees Inter-University Psychometric Entrance Test
This section contains 20 questions. FORMULA PAGE The time allotted is 20 minutes. This section consists of questions and problems involving Quantitative Reasoning. Each question is followed by four possible responses. Choose the correct answer and mark its number in the appropriate place on the answer sheet. Note: The words appearing against a gray background are translated into several languages at the bottom of each page. General Comments about the Quantitative Reasoning Section
- The figures accompanying some of the problems are provided to help solve the problems, but are not necessarily drawn to scale. Therefore, do not rely on the figures alone to deduce line length, angle size, and so forth.
- If a line in a figure appears to be straight, you may assume that it is in fact a straight line.
- When a geometric term (side, radius, area, volume, etc.) appears in a problem, it refers to a term whose value is greater than 0, unless stated otherwise.
- When a (a > 0) appears in a problem, it refers to the positive root of a.
- "0" is neither a positive nor a negative number.
- "0" is an even number.
- "1" is not a prime number.
Formulas
1. Percentages: a% of x is equal to 100 a^ $x 2. Exponents: For every a that does not equal 0, and for any two integers n and m -
a. a a
1 = n −n
b. am^ +^ n^ = am^ · an
c. a m ma
n (^) n = _^ i^ (0 < a, 0 < m)
d. an^ ·^ m^ = (an)m
3. Contracted Multiplication Formulas: (a ± b)^2 = a^2 ± 2ab + b^2 (a + b)(a – b) = a^2 – b^2 4. Distance Problems: distancetime^ = speed (rate) 5. Work Problems: amount of worktime^ = output (rate) 6. Factorials: n! = n(n – 1)(n – 2) · ... · 2 · 1
7. Proportions: If AD || BE || CF
then (^) DEAB^ = BCEFand (^) ACAB^ =DEDF
8. Triangles: a. The area of a triangle with base of length a and altitude to the base of length h is a h 2 $ b. Pythagorean Theorem: In any right triangle ABC, as in the figure, the following always holds true: AC^2 = AB^2 + BC^2 c. In any right triangle whose angles measure 30°, 60°, 90°, the length of the leg opposite the 30° angle is equal to half the length of the hypotenuse. 9. The area of a rectangle of length a and width b is a · b
a
b h
r r
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c a b
A
h
r
B C
D E F h a A
B C
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A
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A
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leg
leg
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A
B C
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a
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A
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A
B C
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10. The area of a trapezoid with one base length a, the other base length b, and altitude h is ]a^ + 2 b g$h 11. The sum of the internal angles of an n-sided polygon is (180n – 360) degrees. In a regular n-sided polygon, each internal angle measures a^180 n^ n^ −^360 k^ =a 180 −^360 nk^ degrees. 12. Circle: a. The area of a circle with radius r is πr^2 (π = 3.14...)
b. The circumference of a circle is 2πr
c. The area of a sector of a circle with a central angle of x° is πr 2 $ 360 x
13. Box (Rectangular Prism), Cube: a. The volume of a box of length a, width b and height c is a · b · c b. The surface area of the box is 2 ab + 2 bc + 2 ac c. In a cube , a = b = c 14. Cylinder: a. The lateral surface area of a cylinder with base radius r and height h is 2 πr · h b. The surface area of the cylinder is 2 πr^2 + 2πr · h = 2πr(r + h) c. The volume of the cylinder is πr 2 · h 15. The volume of a cone with base radius r and height h is πr^3 $h
2
16. The volume of a pyramid with base
area S and height h is S^3^ $ h
a
b h
r r
x°
c a b
A
h
r
B C
D E F h a A
B C
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B C
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b
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h ספרדית רוסית משולב צרפתית a
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a
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h r
Quantitative Reasoning
38
Prime number: A positive integer that is evenly divisible by only two numbers – itself and the number 1. For example, 13 is a prime number because it is evenly divisible by only 1 and 13. Note: 1 is not defined as a prime number.
Opposite numbers (additive inverse): A pair of numbers whose sum equals zero. For example, 4 and (-4) are opposite numbers. In general, a and (-a) are opposite numbers (a + (-a) = 0). In other words, (-a) is the opposite number of a.
Reciprocals (multiplicative inverse:) A pair of numbers whose product is equal to 1. For example, 3 and 31 are reciprocals, as are 72 and 27. In general, for a , b ≠ 0:
a and (^) a^1 are reciprocals aa $ (^) a^1 = 1 k. We can also say that (^) a^1 is the
reciprocal of a.
b
a (^) and a
b (^) are reciprocals 1 b
a a b $^ b= l, or in other words,^ a
b (^) is the reciprocal
of (^) ba^.
Absolute value: If x > 0, then (^) | x (^) | = x.
If 0 > x, then (^) | x (^) | = -x.
|^0 | = 0.
ARITHMETICAL OPERATIONS WITH EVEN AND ODD NUMBERS
even + even = even odd + odd = even
odd + even = odd
even –^ even = even odd –^ odd = even even – odd = odd odd – even = odd
even × even = even odd × odd = odd odd × even = even
Quantitative Reasoning
There are no similar rules for division. The quotient of two even numbers may be odd b^26 = 3 l,
even a^24 = 2 k^ , or a non-integer b^46 = 1 21 l.
FACTORS (DIVISORS) AND MULTIPLES
A factor (divisor) of a positive integer is any positive integer that divides it evenly (that is, without a remainder). For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
A common factor of x and y is a number that is a factor of x and also a factor of y. For example, 6 is a common factor of 24 and 30.
A prime factor is a factor that is also a prime number. For example, the prime factors of 24 are 2 and 3. Any positive integer (greater than 1) can be written as the product of prime factors. For
example, 24 = 3 · 2 · 2 · 2 = 3 · 2^3
A multiple of an integer x is any integer that is evenly divisible by x. For example, 16, 32, and 88 are multiples of 8.
When the word "divisible" appears in a question, it means "evenly divisible" or "divisible without a remainder."
MATHEMATICAL OPERATIONS WITH FRACTIONS
Reduction
When the numerator and denominator of a fraction have a common factor, each of them can be divided by that common factor. The resulting fraction, which has a smaller numerator and
denominator, equals the original fraction. For example, if we divide the numerator and the
denominator of 1216 by 4, the result is 34 , b 1216 = 34 l.
Multiplication
To multiply two fractions, multiply the numerators by each other and the denominators by each other.
Example: 32 $ 75 = (^) 3 72 5$$^ = 2110
Division
To divide a number by a fraction, multiply the number by the reciprocal of the divisor.
Example: 8
3
5
2 5
2 3
8 5 3
2 8 15
= $ = =^16
To multiply or divide an integer by a fraction, the integer can be regarded as a fraction whose
denominator is 1.
Example: 2 = 12
Guide For Examinees Inter-University Psychometric Entrance Test
EXAMPLE
The price of an item dropped from 15 shekels to 12 shekels. By what percentage did the price drop? In this example, the change in the price of an item is given, and you are asked to calculate this change as a percentage. The difference in the price is 3 shekels out of 15 shekels. You have to calculate what percent of 15 is 3.
Convert the question into a mathematical expression: 100 a $ 15 = 3. Solve the equation for a: a = 3 100 15 $^ = 20
Thus, the price dropped by 20%.
RATIO
The ratio of x to y is written as x : y.
EXAMPLE
The ratio between the number of pairs of socks and the number of shirts that Eli has is 3 : 2. In other words, for every 3 pairs of socks, Eli has 2 shirts. Stating it differently, the number of socks that Eli has is 23 greater than the number of shirts that he has.
MEAN (AVERAGE)
The arithmetic mean (average) of a set of numerical values is the sum of the values divided by the number of values in the set.
When the word "average" appears in a question, it refers to the arithmetic mean.
For example, the average of the set of values 1, 3, 5, 10, and 21 is 8 because
5
1 3 5 10 21 5
If the average of a set of numerical values is given, their sum can be calculated by multiplying the average by the number of values.
EXAMPLE
Danny bought 5 items whose average price is 10 shekels. How much did Danny pay for all of the items? In this question we are asked to find the sum based on the average. If we multiply the average by the number of items, we will obtain 10 · 5 = 50. Thus, Danny paid a total of 50 shekels for all of the items which he bought.
A weighted average is an average that takes into account the relative weight of each of the values in a set.
Guide For Examinees Inter-University Psychometric Entrance Test
EXAMPLE
Robert's score on the midterm exam was 75, and his score on the final exam was 90. If the weight of the final exam is twice that of the midterm exam, what is Robert's final grade in the course? The set of values in this case is 75 and 90, but each has a different weight in Robert's final grade for the course. The score of 75 has a weight of 1, while the score of 90 has a weight of 2. To calculate the weighted average, multiply each score by the weight assigned to it, and divide by
the sum of the weights: 1 75$^1 ++^2 2 90$^ = 85. Thus, Robert's final grade in the course is 85. This calculation is identical to the calculation of a simple average of the three numbers 75, 90 and 90.
POWERS AND ROOTS
Raising a number to the nth power (when n is a positive integer) means multiplying it by itself n times: a n^ = a 1 4 $ 4 ... 2 $ a a 44 $ 3. n times For example: (-3)^3 = (-3)(-3)(-3) = -27.
The expression an^ is called a power; n is called the exponent; and a is called the base.
Any number other than zero, raised to the 0th power, equals 1. Thus, for any a ≠ 0, a^0 = 1. A power with a negative exponent is defined as the power obtained by raising the reciprocal of the
base to the opposite power: a n^ a^1
n
- (^) =a k. Example: 2 3 21 21 21 21 81 3
- (^) = (^) a k = $ $ =
The nth^ root of a positive number a, expressed as n^ a, is the positive number b, which if raised to
the nth^ power, will give a. In other words, n^ a = bbecause bn^ = a. For example, 4 81 = 3 because
34 = 81. When the root is not specified, a 2nd-order root is intended. A 2nd-order root is also called a square root. For example, 81 = 281 = 9. A root can also be expressed as a power in which the exponent is a fraction. This fraction is the
reciprocal of the order of the root: n^ a = a n^0
Now let us consider a multi-stage experiment with a set of n items, out of which an item is selected at random r times. Each selection of an item from the set constitutes a stage in the experiment, so that the experiment has a total of r stages. The number of possible results in each of the r stages depends on the sampling method by which an item is selected. The total number of possible results of the entire experiment is the product of the number of possible results obtained in each of the r stages. Each possible result in the experiment is referred to as a sample.
There are four basic types of multistage experiments. They are designated by the sampling method used: whether or not the order of sampling matters (called ordered and unordered) and whether or not the sampled item is returned to the original set (called with replacement and without replacement).
Ordered Samples with Replacement
Sampling method: The sampled item is immediately returned to the set and the order in which the items are sampled matters. Note: In this sampling method, an item may be sampled more than once. Number of possible results: The number of possible results in each stage is n. Thus, the number of possible results of all r stages – that is, of the entire experiment – is n · n · ... · n = nr.
The number of ordered samples with replacement is nr.
EXAMPLE
A box contains 9 balls, numbered 1 through 9. One ball is removed at random from the box and replaced, and this process is repeated two more times. The numbers on the balls that are removed from the box are written down in the order in which they are removed, forming a three- digit number. How many different three-digit numbers can be obtained in this way? In this question, the order in which the results are obtained is important. For example, if balls numbered 3, 8, and 3 are removed, in that order, the number 383 is obtained; but if the order in which they are removed is 3, 3, and 8, the result is a different number – 338. There are 3 stages in this experiment, and the number of possible results in each stage is 9. Thus, the number of possible results of the entire experiment is 9^3 = 729. In other words, 729 different three-digit numbers can be obtained.
Ordered Samples Without Replacement
Sampling method: The sampled item is not returned to the set after being sampled, and the order in which the items are sampled matters. Number of possible results: The number of possible results in the first stage is n; the number of possible results in the second stage is n – 1 (because the item that was sampled in the first stage is not returned, and only n – 1 items remain to be sampled); and so on, until the last stage, stage r, in which the number of possible results is n – r + 1. Thus, the number of possible results of the entire experiment is n · (n – 1) · ... · (n – r + 1). The number of ordered samples without replacement is n · (n – 1) · ... · (n – r + 1).
Quantitative Reasoning
EXAMPLE
A box contains 9 balls, numbered 1 through 9. Three balls are removed at random from the box, one after another, and are not replaced. The numbers on the balls removed from the box are written down in the order in which they are removed, forming a three-digit number. How many different three-digit numbers can be obtained in this way? In this experiment, too, the order in which the balls are removed is important, but unlike the previous example, in this experiment a ball that is removed from the box is not returned. Thus, the number of possible results in the first stage is 9, in the second stage, 8, and in the third stage,
- The number of possible results of the entire experiment is 9 · 8 · 7 = 504. In other words, 504 different three-digit numbers can be obtained.
Arrangements (Permutations) of an Ordered Sample
When creating an ordered sample without replacement out of all n items in a set (that is, if r = n), each possible result describes the arrangement of the items – which item is first, which is second, and so on. The question is: How many possible arrangements are there?
If we substitute r = n in the formula for obtaining the number of ordered samples without
replacement, we obtain n · (n – 1) · ... · 2 · 1. This number is called "n factorial" and is written as n!.
The number of possible arrangements of n items is n!.
EXAMPLE
A grandmother, mother, and daughter wish to arrange themselves in a row in order to be photographed. How many different ways can they arrange themselves? Let us regard the person on the right as the first in the set, the person in the middle as the second, and the person on the left as the third. The question is then: How many possible arrangements are there of the grandmother, mother and daughter? The grandmother, mother, and daughter can be considered a set of 3 items. Thus, the number of possible arrangements for this set is 3! = 3 · 2 · 1 = 6. The possible arrangements are: grandmother-mother-daughter, grandmother- daughter-mother, mother-grandmother-daughter, mother-daughter-grandmother, daughter- grandmother-mother, daughter-mother-grandmother.
Unordered Samples
Sampling method: The sampled item is not returned to the set after being sampled, and the order in which the items are sampled does not matter.
When the order does not matter, all samples containing the same r items (only the sampling order is different in each sample) are regarded as the same result. The number of samples of this type is actually the number of arrangements of the r items, that is, r!.
To calculate the possible number of results in an unordered sample, calculate the number of possible results as if the order matters, and divide it by the number of arrangements of the r items.
The number of unordered samples = the number of ordered samples without replacement the number of arrangements in the sample
= n^ ·^ (n^ –^ 1)^ ·^ ... r!^ ·^ (n^ –^ r^ +^ 1)
Guide For Examinees Inter-University Psychometric Entrance Test
EXAMPLE
Experiment: The tossing of a fair die. Event: The result is less than 4. Results of this event: the numbers 1, 2, and 3.
Probability of the event: 63 = 21.
EXAMPLE
Experiment: The removal of a ball from a box containing 5 white balls and 5 black balls. Event: The removal of a black ball.
Probability of the event: the number of black balls^ = 105 = 12. the total number of balls in the box
The probability that two events will occur
When two events occur at the same time or one after another, there are two possible scenarios:
A. The events are independent , that is, the probability of one event occurring is not affected by the occurrence of the other event.
B. The events are dependent , that is, the probability of one event is affected by the occurrence of another event. In other words, the probability of a particular event occurring after (or given that) another event has occurred, is different from the probability of that particular event occurring independent of the other event.
EXAMPLE
There are 10 balls in a box, 5 white and 5 black. Two balls are removed from the box, one after another. The first ball that is removed is black. What is the probability that the second ball that is removed is also black?
(a) The first ball is returned to the box.
Since the first ball is returned to the box, there is no change either in the total number of balls in the box or in the number of black balls.
The probability of removing a second black ball is 105 = 21 and is equal to the probability that the first ball that was removed is black.
The fact that a black ball was removed the first time does not change the probability of removing a black ball the second time. In other words, the two events are independent.
(b) The first ball is not returned to the box.
After removing a black ball, a total of 9 balls remain in the box, of which 4 are black. Thus, the probability of removing a second black ball is 94.
This probability is different from the probability of removing a black ball the first time. Thus, the second event is dependent on the first event.
Guide For Examinees Inter-University Psychometric Entrance Test
The probability of two independent events occurring (in parallel or one after another) is equal to the product of the probabilities of each individual event occurring.
EXAMPLE Experiment: The tossing of two fair dice, one red and the other yellow. Event A: obtaining a number that is less than 3 on the red die. The probability of event A is
6
2 3
Event B: obtaining an even number on the yellow die. The probability of event B is 63 = 12. Since the result of tossing one die does not affect the probability of the result obtained by tossing the other die, event A and event B are independent events. The probability of both event A and event B occurring (together) is thus, the probability (^) × the probability of event A of event B
, that is, 31 $ 21 = 61.
Let us define two dependent events, A and B (in any given experiment).
The probability of event B occurring given that event A has occurred is:
the number of results common to both A and B the number of results of A
EXAMPLE
Experiment: The tossing of a die. What is the probability of obtaining a result that is less than 4 if we know that the result obtained was an even number? Event A: an even result is obtained. Event B: a result that is less than 4 is obtained.
We will rephrase the question in terms of the events: What is the probability of B, given that we know that A occurred? Event A has 3 results: 2, 4, and 6. Event B has 3 results: 1, 2, and 3. But, if we know that event A occurred, there is only one possible result for B – 2. In other words, the result 2 is the only result that is common to both A and B. Thus, the probability of B, given that we know that A occurred is 31. This probability is different from the probability of B occurring (without our knowing anything about A), which is equal to 21.
Distance Problems: Distance, Speed (Rate), Time
The speed (rate) at which an object moves is the distance that the object covers in a unit of time. The formula for the relationship between the speed, the distance the object covers, and the amount of time it requires to cover that distance is:
v = st where v = speed (rate) s = distance t = time
Quantitative Reasoning
EXAMPLE
A builder can finish building one wall in 3 hours. How many hours would be needed for two builders working at the same rate to finish building 5 walls? We are given the amount of work of one builder (1 wall), and the amount of time he spent working (3 hours). Therefore his output is 13 of a wall in an hour. Since the question involves
two builders, the output of the two of them together is 2 $ 31 = 32 walls per hour.
We are also given the amount of work which the two builders must do – 5 walls. We can
therefore calculate the amount of time they will need: t 5 7 3
2
5 2
3 2
15 2
= = $ = =^1. Thus, they will
need 7 21 hours to build the walls.
Quantitative Reasoning
PARALLEL (STRAIGHT) LINES
Parallel lines that intersect any two lines divide those lines into segments that are proportional in length. Thus, in the figure, (^) ca^ = (^) db , (^) ba^ = (^) dc and (^) a a +^ b=c c+ d. Other relationships between the segments can be deduced based on the above relationships.
ANGLES
A right angle is a 90° angle. In all of the figures, right angles are marked by
a
b
h
r r
x°
c a b
A
h
r
B C
D E F h a A
B C
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יתר
A
B C
ﻗﺎﺋﻢ ﻗﺎﺋﻢ
وﺗﺮ
A
B C
hypoténuse
côté
côté
משולב צרפתית
A
B C
hypotenuse
leg
leg
רוסית
A
B C
ubgjntyepf rfntn rfntn
ספרדית
A
B C
hipotenusa
cateto
cateto
h r
An acute angle is an angle that is less than 90°. An obtuse angle is an angle that is greater than 90°. A straight angle is a 180° angle.
Adjacent Supplementary Angles
The two angles that are formed between a straight line and a ray that extends from a point on the straight line are called adjacent supplementary angles. Together they form a straight angle and their sum therefore equals 180°. For example, in the figure, x and y are adjacent supplementary angles; thus, x + y = 180°.
Vertical Angles
When two straight lines intersect, they form four angles. Each pair of non-adjacent angles are called vertical angles and they are equal in size. For example, in the figure, x and z are vertical angles, as are w and y. Therefore, x = z and w = y.
When two parallel lines are intersected by another line (called a transversal), eight angles are formed. In the figure, these angles are designated a, b, c, d, e, f, g, and h.
Corresponding Angles
Angles located on the same side of the transversal and on the same side of the parallel lines are referred to as corresponding angles. Corresponding angles are equal. Thus, in the figure, a = e , b = f , c = g , d = h.
Alternate Angles
Angles located on opposite sides of the transversal and on opposite sides of the parallel lines are called alternate angles. Alternate angles are equal. Thus, in the figure, a = h , b = g , c = f , d = e.
a c
b d
c
e f g h
C
A
B
h
α
δ β^ γ
A
B C
p a b
q
c d
e f g h
A
60° B C
60°
60°
B C
A
h
C
A
B D
y w z
x
a c
b d
a b c d
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a c
b d
a b c d
e f g h
y (^) x
p a^ b
q
c d
e
Guide For Examinees Inter-University Psychometric Entrance Test
53
Congruent Triangles
Two geometric figures are congruent if one of them can be placed on the other in such a way that the two coincide. Congruent triangles are one example of congruent geometric figures. In congruent triangles, the corresponding sides and angles are equal.
For example, if triangles ABC and DEF in the figure are congruent, then their corresponding sides are equal: AB = DE, BC = EF, and AC = DF, and their corresponding angles are equal: a = d, b = τ, and g = ε.
Each of the following four theorems enables us to deduce that two triangles are congruent:
(a) Two triangles are congruent if two sides of one triangle equal the two corresponding sides of the other triangle and the angle between these sides in one triangle equals the corresponding angle in the other triangle (Side-Angle-Side – SAS). For example, if AB = DE, AC = DF, and a = d, then the two triangles in the figure are congruent.
(b) Two triangles are congruent if two angles of one triangle equal the two corresponding angles of another triangle, and the length of the side between these angles in one triangle equals the length of the corresponding side in the other triangle (Angle-Side-Angle – ASA). For example, if a = d, b = τ, and AB = DE, then the two triangles in the figure are congruent.
(c) Two triangles are congruent if the three sides of one triangle equal the three sides of the other triangle (Side-Side-Side – SSS).
(d) Two triangles are congruent if two sides of one triangle equal the corresponding two sides of the other triangle, and the angle opposite the longer of the two sides of one triangle is equal to the corresponding angle in the other triangle (Side-Side-Angle – SSA). For example, the triangles in the figure above are congruent if AB > AC and DE > DF; and AB = DE, AC = DF, and g = ε.
Similar Triangles
Two triangles are similar if the three angles of one triangle are equal to the three angles of the other triangle.
In similar triangles, the ratio between any two sides of one triangle is the same as the ratio between the corresponding two sides of the other triangle.
For example, in the figure, triangles ABC and DEF are similar.
Therefore, (^) ACAB = (^) DFDE.
It also follows that (^) DEAB = (^) DFAC= (^) EFBC.
Congruent triangles are necessarily also similar triangles.
a c
b d
A
β γ
α
D
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δ
E F
B C
C
A
B
h
A
B
ניצב
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80°
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h
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a c
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a b c d
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a
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A
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p a b
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80° 40° (^) 60°
B C
A
h
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A
B D
a
a
45°
45°
a 2
Guide For Examinees Inter-University Psychometric Entrance Test
TYPES OF TRIANGLES
An equilateral triangle is a triangle whose sides are all of equal length. For example, in the figure, AB = BC = AC. In a triangle of this type, all of the angles are also equal (60°).
If the length of the side of such a triangle is a, then its altitude is a $ 23
and its area is a^2 $ 43.
An isosceles triangle is a triangle with two sides of equal length. For example, in the figure, AB = AC. The third side of an isosceles triangle is called the base. The two angles opposite the equal sides are equal. For example, in the figure, b = g.
An acute triangle is a triangle in which all the angles are acute.
An obtuse triangle is a triangle with one obtuse angle.
A right triangle is a triangle with one right angle (90°). The side opposite the right angle (side AC in the figure) is called the hypotenuse , and the other two sides are called legs (sides AB and BC in the figure). According to the Pythagorean theorem, in a right triangle the square of the hypotenuse is equal to the sum of the squares of the legs. For example, in the figure, AC^2 = AB^2 + BC^2. This formula can be used to find the length of any side if the lengths of the other two sides are given.
In a right triangle whose angles measure 30°, 60° and 90°, the length of the leg opposite the 30° angle equals half the length of the hypotenuse. For example, in the figure, the length of the hypotenuse is 2a. Therefore, the length of the leg opposite the 30° angle is a. It follows from the Pythagorean theorem that the length of the leg opposite the 60° angle is a 3.
In an isosceles right triangle, the angles measure 45°, 45°, and 90°, the two legs are of equal length, and the length of the hypotenuse is 2 times greater than the length of the legs (based on the Pythagorean theorem). For example, in the figure, the length of each leg is a, and therefore the length of the hypotenuse is a 2.
B C
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h
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A
B D
y w z
x
a c
b d
a b c d
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y (^) x
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β γ
α
D
τ ε
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A
B β^ C
γ
A
h
a
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30°
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a 3
A
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hypoténuse
côté
côté
c
צרפתית
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α
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p a b
q
c d
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h
a
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y w z
x
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b d
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β γ
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A
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A
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h
a
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a (^3)
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hypoténuse
côté
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hipotenusa
cateto
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צרפתית
ספרדית
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leg
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אנגלית
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יתר
ניצב
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A
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B ﻗﺎﺋﻢ C
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α
δ β^ γ
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p a b
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h
a
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y w z
x
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b d
a b c d
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y (^) x
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β γ
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β γ
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h
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p a b
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y w z
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b d
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β γ
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h
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p a b
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h
a
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a 2
Quantitative Reasoning
56
