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.

Triangle ABC

m∠A (Common angle)

m∠B (Right angle)

Triangle ADE

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 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

⋅ 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.

Triangle ABE

m∠E (Common angle)

m∠B (Right angle)

Triangle ADE

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 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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