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Classifying and Constructing Triangles: Understanding Triangle Properties and Theorems, Study notes of Analytical Geometry and Calculus

Yale UniversityAnalytical Geometry and Calculus

A learning unit from siyavula, focusing on classifying triangles based on their angles and sides, discovering important theorems about triangles, and constructing triangles using compasses, protractor, pencil, and ruler. Students will learn to identify acute-angled, right-angled, and obtuse-angled triangles, as well as isosceles, equilateral, and scalene triangles. They will also learn to apply theorems such as the sum of interior angles and the exterior angle of a triangle. The unit includes practical exercises to help students understand the concepts.

Typology: Study notes

2011/2012

Uploaded on 10/19/2012

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Connexions module: m31148 1
Classifying and constructing
triangles
∗
Siyavula Uploaders
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
1 MATHEMATICS
2 Grade 8
3 RATIONAL NUMBERS, CIRCLES AND TRIANGLES
4 Module 14
5 CLASSIFYING AND CONSTRUCTING TRIANGLES
5.1 ACTIVITY 1
5.2 Classifying triangles, discovering important theorems about triangles and constructing
triangles
5.3 [LO 3.1, 3.3, 3.4, 4.2.1]
•
By the end of this learning unit, you will be able to do the following:
•
understand how important the use of triangles is in everyday situations;
•
explain how to nd the unknown sides of a right-angled triangle (Pythagoras);
•
calculate the area of a triangle;
•
enjoy the action in geometry;
•
use mathematical language to convey mathematical ideas, concepts, generalisations and mental pro-
cesses.
1. When you classify triangles you can do it according to the angles or according to the sides.
1.1 Classication on the basis of the angles of a triangle:Are you able to complete the following?
a) Acute-angled triangles are triangles with
b) Right-angled triangles have
c) Obtuse-angled triangles have
1.2 Classication on the basis of the sides of the triangle:Are you able to complete the following?
a) An isosceles triangle has
b) An equilateral triangle has
∗
Version 1.1: Aug 8, 2009 3:36 pm GMT-5
†
http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m31148/1.1/
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pf4
pf5
pf8
pf9
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Download Classifying and Constructing Triangles: Understanding Triangle Properties and Theorems and more Study notes Analytical Geometry and Calculus in PDF only on Docsity!

Classifying and constructing

triangles

Siyavula Uploaders

This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †

1 MATHEMATICS

2 Grade 8

3 RATIONAL NUMBERS, CIRCLES AND TRIANGLES

4 Module 14

5 CLASSIFYING AND CONSTRUCTING TRIANGLES

5.1 ACTIVITY 1

5.2 Classifying triangles, discovering important theorems about triangles and constructing triangles

5.3 [LO 3.1, 3.3, 3.4, 4.2.1]

  • By the end of this learning unit, you will be able to do the following:
  • understand how important the use of triangles is in everyday situations;
  • explain how to nd the unknown sides of a right-angled triangle (Pythagoras);
  • calculate the area of a triangle;
  • enjoy the action in geometry;
  • use mathematical language to convey mathematical ideas, concepts, generalisations and mental pro- cesses.
  1. When you classify triangles you can do it according to the angles or according to the sides. 1.1 Classication on the basis of the angles of a triangle:Are you able to complete the following? a) Acute-angled triangles are triangles with b) Right-angled triangles have c) Obtuse-angled triangles have 1.2 Classication on the basis of the sides of the triangle:Are you able to complete the following? a) An isosceles triangle has b) An equilateral triangle has ∗Version 1.1: Aug 8, 2009 3:36 pm GMT- †http://creativecommons.org/licenses/by/3.0/

c) A scalene triangle's

  1. Are you able to complete the following theorems about triangles? Use a sketch to illustrate each of the theorems graphically. THEOREM 1:
  • The sum of the interior angles of any triangle is.........................

Sketch: THEOREM 2:

  • The exterior angle of a triangle is

Sketch:

  1. Constructing triangles:
  • Equipment: compasses, protractor, pencil and ruler

Remember this:

  • Begin by drawing a rough sketch of the possible appearance.
  • Begin by drawing the base line.

3.1 Construct ∆PQR with PQ = 7 cm, PR = 5 cm and

Θ P = 70 ◦. a) Sketch: b) Measure the following:

  1. QR = ........ 2.

Θ R = ........ 3.

Θ Q (^) = ........ 4.

Θ P +

Θ Q (^) +

Θ R = ........ 3.2 Construct ∆KLM , an equilateral triangle. KM = 40 mm, KL=LM and

Θ K = 75^ ◦.Indicate the sizes of all the angles in your sketch. Sketch:

6 ACTIVITY 2

7 Discovering the Pythagorean theorem of Pythagoras and calculating unknown

sides with the help of this theorem

8 [LO 4.2.1, 4.8, 4.9, 4.10]

  • The following could be done in groups.

Practical exercise: Making you own tangram.

  1. Cut out a cardboard square (10 cm x 10 cm).
  2. Draw both diagonals, because they form part of the bases of some gures.
  3. Divide the square in such a way that the complete gure consists of the following: 3.1 two large equilateral triangles with bases of 10 cm in length; 3.2 two smaller equilateral triangles, each with base 5 cm in length; 3.3 one medium equilateral triangle with adjacent sides 5 cm in length; 3.4 one square with diagonals of 5cm; 3.5 one parallelogram with opposite sides of 5 cm.
  • Make two of these. Cut along all the lines so that you will have two sets of the above shapes.

Figure 5

  1. Do the calculations to determine whether the following is a right-angled triangle or not: 12.1 ∆DEF with DE = 8 cm, EF = 10 cm, DF = 6 cm
  2. AREA OF TRIANGLES 13.1 Construct rectangle ABCD with AB = 45 mm and AD = 25 mm on a sheet of paper and cut it out. Draw diagonal AC. 13.2 Calculate the area of rectangle ABCD. 13.3 Cut out ∆ABC. What is the area of ∆ABC?Paste it here.
  • Area of ∆ABC = ................. mm^2

13.4 Are you able to develop a formula for determining the area any triangle? Write it here: 13.5 Calculate the area of ∆ABC.

Figure 6

13.6 In the gure SQ = 15 cm, QR = 7 cm and PR = 9 cm. Important: Provide all necessary information on your sketch. Check to see what you may need to complete the instructions fully.

14. Calculate the length of the unknown sides of each of the following:

Figure 11

  1. Playing in a park is a necessary aspect of the development of a child.
  • You have been asked to supply slides. The problem that is involved requires calculating the length of the poles that are needed. Make use of the knowledge that you have accumulated to supply a plan to erect the slides.

Figure 12

The following is required: 15.1 a sketch 15.2 a scale, e.g. 1 cm = 1 m 15.3 Calculations must be completed fully.

9 Assessment

LO 3

continued on next page

b) 3 even sides c) sides dier in length

  1. The sum of the interior angles of any triangle is 180º ACTIVITY 2
  2. r^2 = p^2 + q^2
  • x^2 = 12^2 + 5^2

= 144 + 25 = 169 ∴x = 13

  • 202 = 8^2 + x^2

x^2 = 400  64 = 336 ∴x 18,3 cm 11.3 ∇ABC: x^2 = 70^2  29^2 = 4 900  841 = 4 059 ∴x 63,7 mm 11.4 y^2 = 4^2 + 3^2 = 16 + 9 = 25 ∴x 9,4cm

  1. DE^2 + DF^2 = 100 = EF^2 ∴DEF right angled (Pythagoras)
  • ½ x b x h
  • BC^2 = 13^2  5^2

= 169  25 = 144 ∴BC = 12 cm Area ABC = ½ x b x h = ½ x 12 x 5 = 30cm^2 13.6 (a) PS^2 = 9^2  8^2 = 81  64 = 17 ∴PS = 4,12 cm Area PSQ = ½ x b x h = ½ x 15 x 4, = 30,9cm^2 13.6 (b) Area PSR = ½ x 8 x 4, = 16,4 cm^2 Area PRQ = area PSQ  PSR = 30,9  16, = 14,5 cm^2 13.7 AC^2 = 12^2 + 8^2 = 208

Area ABCD = area ABC + area ACD

  • Figure
  • Figure
      • Figure
    • Figure
      • ∴AC 14,
      • AD^2 = 16^2  14,4
      • = 256  207,
      • = 48,
      • ∴AD = 6,
      • = 48 + 50, = (½ x 12 x 8) + (6,97 x 14,4 x ½)
    • • a^2 = 8^2  = 98,18 square units
  • = - ∴a 3, - b^2 = (3,9)^2 + - = 15,21 + - = 31, - ∴b 5,
  • y^2 = 36^2  • x = 18 (radius) - = 1 296  - = - ∴y = 33,
    • • UV 2 = 12^2 
  • = - ∴UV = 9, - VS^2 = 14^2 + ( 9,8) - = 196 + - = - ∴VS = 17, - y^2 = ( 17,1)^2 + - = 291 + - = - ∴y = 17,