FINDING MISSING MEASURES IN SIMILAR TRIANGLES WORKSHEET

Problem 1 : 

While playing tennis, David is 12 meters from the net, which is 0.9 meter high. He needs to hit the ball so that it just clears the net and lands 6 meters beyond the base of the net. At what height should Matt hit the tennis ball ?

Solution :

Step 1 :

Draw an appropriate diagram to the given information

In the above diagram, 

BC  =  Height of the net

DE  =  Height of ball when hit

A  =  Point at where the ball lands

Step 2 : 

Let us compare two corresponding angles of triangle ABC and ADE. 

Because two angles in one triangle are congruent to two angles in the other triangle, the two triangles are similar.

Step 3 : 

Since the triangle ABC and ADE are similar triangles,  corresponding side lengths are proportional.

So, we have

AD / AB  =  DE / BC

(AB + BD) / AB  =  DE / BC

Substitute the lengths from the figure.

(6 + 12) / 6  =  h / 0.9

18 / 6  =  h / 0.9

3  =  h / 0.9

Multiply both sides by 0.9

3 ⋅ 0.9  =  (h/0.9) ⋅ 9

2.7  =  h

So, David should hit the ball at a height of 2.7 meters.

Problem 2 : 

Jose is building a wheelchair ramp that is 24 feet long and 2 feet high. She needs to install a vertical support piece 8 feet from the end of the ramp. What is the length of the support piece in inches ?

Solution :

Step 1 :

Draw an appropriate diagram to the given information

In the above diagram, 

AB  =  Height of the chair

CD  =  Height of the support piece

E  =  End of the ramp

Step 2 : 

Let us compare two corresponding angles of triangle ABE and CDE. 

Because two angles in one triangle are congruent to two angles in the other triangle, the two triangles are similar.

Step 3 : 

Since the triangle ABE and ADE are similar triangles,  corresponding side lengths are proportional.

So, we have

DE / BE  =  CD / AB

Substitute the lengths from the figure.

8 / 24  =  h / 2

1 / 3  =  h / 2

Multiply both sides by 2.

(1/3) ⋅ 2  =  (h/2) ⋅ 2

2/3 ft  =  h

Step 4 : 

Convert feet into inches.

Because 1 ft = 12 inches, we have to multiply by 12 to convert feet into inches.   

h  =  2/3 ft

h  =  (2/3) ⋅ 12 inches

h  =  8 inches

So, the length of the support piece is 8 inches.

Problem 3 : 

To find the length of a pond, a surveyor took some measurements. She recorded them on this diagram. What is the length of the pond

Solution :

In triangle KLP and PMN

∠LPK = ∠MPN = 90

∠KLP = ∠PMN = 76

Using AA, the triangles are similar.

Let x be the length of the pond.

3/5 = 12/x

3x = 12(5)

3x = 60

x = 60/3

x = 20 m

So, the length of the pond is 20 m.

Problem 4 : 

Movie screens often have an aspect ratio of 16 by 9. This means that for every 16 ft of width along the base of the screen, there is 9 ft of height. The width of the screen at the Airport Cinemas is about 115 ft. The screen has a 16:9 aspect ratio. About how tall is the screen?

Solution :

When width of the screen = 16 ft

height of the scresn = 9 ft

When width of the screen is 115 ft, let x be the height of the screen.

16 : 9 = 115 : x

16/9 = 115/x

16x = 115(9)

x = 115(9)/16

x = 64.68

Approximately 64.7 ft.

Problem 5 : 

Yolanda uses the shadow method to estimate the height of a flagpole. Her height of 5 feet casts a 4 foot shadow. At the same time, she finds that the school’s flagpole casts a shadow that is 21 feet long. Sketch a diagram and use a proportion to find the height of the flagpole?

Solution :

When height = 5 feet

length of shadow = 4 foot

Let x be the height of the flagpole

The flagpole which has length of shadow as 21 ft it creates the height as x.

5/4 = x/21

Doing cross multiplication, we get

5(21) = 4x

x = 105/4

x = 26.25 ft

So, the height of the flagpole is 26.25 ft

Problem 6 : 

A tree casts a shadow that is 24 feet long. A person who is 5 feet tall is standing in front of the tree, and his shadow is 8 feet long. Approximately how tall is the tree?

Solution :

Let h be the height of the tree.

Comparing the corresponding sides

h/5 = 24/8

h = 3(5)

h = 15 ft

So, the height of the tree is 15 ft.

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