SIMILAR RIGHT TRIANGLES WORKSHEET

Problem 1 :

Find the value of x in the diagram shown below.

Problem 2 :

Find the value of y in the diagram shown below.

Problem 3 :

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.

b.  Find the height h of the roof.

Problem 4 :

To estimate the height of a monorail track, your friend holds a cardboard square at eye level. Your friend lines up the top edge of the square with the track and the bottom edge with the ground. You measure the distance from the ground to your friend’s eye and the distance from your friend to the track.

In the diagram shown below, XY = h - 5.75 is the difference between the track height h and your friend’s eye level. Find the height of the track h.

1. Answer :

Apply Geometric Mean Theorem 1.

x/3 = 6/x

Apply cross product property.

x² = 18

Solve for x.

x = √18

x = √(3 ⋅ 3 ⋅ 2)

x = 3√2

2. Answer :

Apply Geometric Mean Theorem 2.

y/2 = (5 + 2)/y

y / 2  =  7 / y

Apply cross product property.

y² = 14

Solve for x.

x = √14

3. Answer :

Part (a) :

We may find it helpful to sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. Mark the congruent angles.

Notice that some sides appear in more than one triangle.

For instance, BC is the hypotenuse in ΔBCA and the shorter leg in ΔBDC.

In the diagram shown above,

ΔBCA ∼ ΔCDA ∼ ΔBDC

Part (b) :

Use the fact that ΔBCA ∼ ΔBDC to write a proportion.

Corresponding side lengths are in proportion :

CA/DC = BC/BD

Substitute.

h/5.5 = 3.1/6.3

Apply cross product property.

6.3h = 3.1(5.5)

Solve for h.

h ≈ 2.7

Hence, the height of the roof is about 2.7 meters.

4. Answer :

Use Geometric Mean Theorem 2 to write a proportion involving XY. Then we can solve for h.

Geometric Mean Theorem 2 :

XY/WY = WY/ZY

Substitute.

(h - 5.75)/16 = 16/5.75

Apply cross product property.

5.75(h - 5.75) = 16²

Distributive property.

5.75h - 33.0625 = 256

Add 33.0625 to each side.

5.75h = 289.0625

Divide each side by 5.75.

h ≈ 50

Hence, the height of the track is about 50 feet.

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