FINDING MISSING MEASURES IN SIMILAR TRIANGLES

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

Example 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.

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.

Example 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 = 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 to inches.

h = (2/3) ⋅ 12 inches

h = 8 inches

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

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