**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.

**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)

**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 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

3 ⋅ 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)

**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 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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