ACT Math : How to find if right triangles are similar

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : Right Triangles

You have two right triangles that are similar.  The base of the first is 6 and the height is 9.  If the base of the second triangle is 20, what is the height of the second triangle?

Possible Answers:

30

23

33

35

25

Correct answer:

30

Explanation:

Similar triangles are proportional.

Base1 / Height1 = Base2 / Height2

6 / 9 = 20 / Height2

Cross multiply  and solve for Height2

6 / 9 = 20 / Height2

6 * Height2=  20 * 9

Height2=  30

Example Question #2 : Right Triangles

A right triangle is defined by the points (1, 1), (1, 5), and (4, 1).  The triangle's sides are enlarged by a factor of 3 to form a new triangle.  What is the area of the new triangle?

Possible Answers:

81 square units

54 square units

None of the answers are correct

36 square units

108 square units

Correct answer:

54 square units

Explanation:

The points define a 3-4-5 right triangle.  Its area is A = 1/2bh = ½(3)(4) = 6.  The scale factor (SF) of the new triangle is 3.  The area of the new triangle is given by Anew = (SF)2 x (Aold) =

32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).

NOTE:  For a volume problem:  Vnew = (SF)3 x (Vold).

Example Question #1 : How To Find If Right Triangles Are Similar

On a flat street, a light pole 36 feet tall casts a shadow that is 9 feet long. At the same time of day, a nearby light pole casts a shadow that is 6 feet long. How many feet tall is the second light pole?

Possible Answers:

Correct answer:

Explanation:

Start by drawing out the light poles and their shadows.

 

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In this case, we end up with two similar triangles. We know that these are similar triangles because the question tells us that these poles are on a flat surface, meaning angle B and angle E are both right angles. Then, because the question states that the shadow cast by both poles are at the same time of day, we know that angles C and F are equivalent. As a result, angles A and D must also be equivalent.

Since these are similar triangles, we can set up proportions for the corresponding sides.

 

 

Now, solve for  by cross-multiplying.

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