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A lesson on solving problems that involve triangle similarity and right triangles. It includes various examples and exercises to help students understand the concepts of similar triangles, mean proportional, and indirect measurement. The document also covers theorems related to right triangles and strategies for arriving at answers.
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Department of Education
Quarter 3 - Module 8: Word Problems: Triangle Similarity and Right Triangle
Z est for P rogress Z eal of P artnership
Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________
Good Day Scout! In the previous module, you learned about triangle similarity theorems/Pythagorean theorem and you have proven each of them. In this module, you will learn to apply those theorems to solve word problems involving triangle similarity and right triangles.
What I Need to Know
The module contains only one lesson: Lesson 9- Solving problems involving triangle similarity and right triangle. After going through this module, you are expected to:
What I Know
Letโs find out how much you already know about this topic. Write only the letter of the choice that you think best answers the question. Write the chosen letter on the space provided for you.
____1. Lance the alien is 5 feet tall. His shadow is 8 feet long. At the same time of day, a treeโs shadow is 32 feet long. What is the height of the tree?
a. 20 feet b. 24 feet c. 29 feet d. 51 feet
Module 8
____ 6. How long is the height of a school flagpole if at a certain time of day, a 5-foot student casts a 3-feet shadow while the length of the shadow cast by the flagpole is 12 ft a. 20 ft. b.18 ft. c. 16 ft. d. 15 ft. ____ 7. The length of the shadow of your one-and-a-half-meter height is 2.4 meters at a certain time in the morning. How high is a tree in your backyard if the length of its shadow is 16 meters? a. 25.6 m b. 10 m c. 38.4 m d. 24 m ____ 8. A flagpole 3 meters tall casts a shadow 5 meters long at the same time that a building nearby casts a shadow 62 meters long. How tall is the building? a. 37.2m b. 35m c. 35.2m d. 40m
____ 9. A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. What is the height h of the roof?
a. h โ 2. b. h โ 4. c. h โ 3. d. h โ 1.
____10. Given in the figure below. What is the value of x?
a. 48 b. 40 c. 58 d. 30
Whatโs In
Direction: State whether or not the following triangles are similar. If not, explain why not. If so, write a similarity statement.
Similar? ___________ Why or why not?_____________________
If so, similarity statement and scale factor: ______________________SF:_________
Similar? ___________ Why or why not?_____________________
If so, similarity statement and scale factor: ______________________SF:_________
Whatโs New
A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. a. Identify the similar triangles.
Mean Proportional (or Geometric Means) appear in two popular theorems regarding right triangles. Before we state these theorems, let's take a look back at a theorem relating to the triangles we will be using:
THEOREM: The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
Then,
THEOREM: The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse
Then, Altitude Rule:
, or THEOREM: The leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.
Then, Leg Rule:
,
Example 1****. Using Indirect Measurement Find the value of the height, h m, in the following diagram at which the tennis ball must be hit so that it will just pass over the net and land 6 meters away from the base of the net.
Solution We draw and label a diagram as shown. Heigth is measured vertically, So, โ EDA is a right angle.
We assume that the net is vertical. ADE and ABC are similar as they are equiangular
โ 0.9 =
h = 2.7 m So, that height at which the ball should be hit is 2.7meters Note: a. Equal angles are marked in the same way in diagrams. b. Two triangles are similar if:
Example 3****.
Find the distance from one side of a riverbank to a tree on the opposite side. Solution. Similar triangles are drawn with right angles, R and Q. S is located by sighting from P to T. By direct measurements, RS = 50m, PQ = 8m and QS = 10m.
RST ~ QSP Then, ๐๐๐๐ ๐๐๐๐ =^
Note: RS = 50m, PQ = 8m and QS = 10m. ๐๐๐๐ 8 =
Now that you have a deeper Understanding of the topic, you are now ready to do the task in the next section.
Whatโs More
Directions: Complete the following solution to find the value of each variable. Answer the questions that follow. Write your answer on the space provided.
ADE and ABC are similar as they are (1.)____________ โ ๐ถ๐ถ๐ถ๐ถ =^
_h = (5.)_____
Questions: a. What will happen if the steps are not followed in solving? _Answer: ______________________________
b. Has your knowledge on proportion helped you in performing the tasks in this activity? _Answer: _____________________________
c. Are the concepts and skills learned in this activity useful to you in the future? how? Answer: _____________________________
Assessment
Directions: Analyze each question and choose the letter of the correct answer. Write the letter of your answer on the space provided before the number.
_____ (^) 1. In a right triangle, what is TRUE about the hypotenuse? a. It is always the longest side. b. It is opposite the acute angle. c. It is always greater than the sum of the other two sides. d. The hypotenuse is always equal to the sum of the other two sides.
____ (^) 2. In a Proportion if ๐๐ ๐๐ = ๐๐ ๐๐ then which of the following statement is not true?
a. ๐๐๐๐ = ๐๐๐๐ b. ๐๐๐๐ = ๐๐๐๐ c. ๐๐+๐๐๐๐ = ๐๐+๐๐๐๐ d. ๐๐+๐๐๐๐ = ๐๐+๐๐๐๐
_____ 3. In the figure, there are three similar right triangles by Right Triangle ProportionalityTheorem. Which triangle is missing in this statement โ HOP ~_______ โOEP?
a. HOE b. OEH c. HOP d. HEO
_____ (^) 4. Which statements is NOT true?
a. The altitude to the hypotenuse is the geometric mean between the segments into which it separates the hypotenuse. b. The leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg. c. The geometric mean is the product of the hypotenuse and the two legs of a right triangle. d. The geometric mean is the square root of the product of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
_____ (^) 5. A flagpole casts a shadow that is 50 feet long. At the same time, you who are 64 inches tall cast a shadow that is 40 inches long. How tall is the flagpole to the nearest foot? a. 12 feet b. 80 feet c. 40 feet d. 140 feet
For questions 6-
Directions: Find the missing length indicated. Match column A with the correct answer on column B. write only the letter of answer on the space provided.
Column A Column B
_____6. a. 48
b. 15
c. 12
d. 27 โ 5
e. 3 โ 7
f. 27 โ 10
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