# USING SIMILAR TRIANGLES TO MEASURE HEIGHT

## About "Using similar triangles to measure height"

Using similar triangles to measure height :

Because corresponding angles are congruent and corresponding sides are proportional in similar triangles, we can use similar triangles to measure height in real-world problems.

## Using similar triangles to measure height - Practice problems

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 triangles ABC and ADE.

Triangle ABC

m∠A (Common angle)

m∠B (Right angle)

m∠A (Common angle)

m∠D (Right angle)

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

Step 3 :

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

So, we have

AD / DB  =  DE / BC

(AB + BD) / DB  =  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

⋅ 0.9  =  (h/0.9) ⋅ 9

2.7  =  h

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

Problem 2 :

The lower cable meets the tree at a height of 6 feet and extends out 16 feet from the base of the tree. If the triangles are similar, how tall is the tree ? Solution :

Step 1 :

Draw an appropriate diagram to the given information In the above diagram,

AB  =  Height of the tree

CD  =  Height at where the lower cable meets the tree

Step 2 :

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

So, we have

AB / DB  =  BC / BE

Substitute the lengths from the figure.

h / 6  =  56 / 16

h / 6  =  7 / 2

Multiply both sides by 6.

(h/6) ⋅ 6  =  (7/2) ⋅ 6

h  =  21

Hence, the height of the tree is 21 ft.

Problem 3 :

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 height 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.

Triangle ABE

m∠E (Common angle)

m∠B (Right angle)

m∠E (Common angle)

m∠D (Right angle)

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

Step 3 :

Since the triangles 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

or

h  =  2/3 ft

Step 4 :

Convert feet into inches.

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

h  =  2/3 ft ----> h  =  (2/3) ⋅ 12 inches

h  =  8 inches

Hence, the height of the support piece is 8 inches. After having gone through the stuff given above, we hope that the students would have understood "Using similar triangles to measure height".

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