6th Grade Math Textbook, Fundamentals

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Problem 2: Equilateral triangle ABChas sides of length 10 cm.
Suppose a point Pis chosen inside the triangle. No matter where
Pis located, the sum of the distances from Pto the three sides
(abcin the diagram at the right) will be the same. What is
this common sum?

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Lesson 13-13 for exercise sets.

A

CB

a

b

P c

10 cm

10 cm 10 cm

Remember:The distance

from a point to a line is the

length of the perpendicular

line segment from the point

to the line.

Read to understand what is being asked.

List the facts and restate the question.

Facts: A point Pis located inside an equilateral triangle

with 10-cm sides.

No matter where Pis located, the sum of the

distances from Pto the three sides will be the same.

Question:What is this common sum?

Select a strategy.

Use the strategy Consider Extreme Cases.

Apply the strategy.

Because the sum of the three distances does not depend on the particular point,

you can put Pin an extreme position to make your work easy. Put Pat vertex A.

This makes two of the distances very easy to find. The distance afrom

Pto the left side is 0 cm. The distance cfrom Pto the right side is 0 cm.

The distance bfrom Pto the base is the length of a leg of a right triangle

with hypotenuse length 10 cm and leg length 5 cm. This distance can be

found by using the Pythagorean Theorem.

b^2  52  102 Use the Pythagorean Theorem.

b^2  25  100 Compute the squares.

b^2  75 Subtract 25 from both sides, and simplify.

b Take the square root of both sides.

b Simplify.

So a b c 0  53 0  53. The sum is 53 cm, or about 8.66 cm.

53

75

A  P

CB

b

10 cm

5 cm

10 cm 10 cm

Check to make sure your answer makes sense.

Draw several equilateral triangles with side lengths of 10 cm.

Choose a different point inside of each. Measure the distance

from the point to each side. Add each set of distances to verify

that the sum is close to 8.66 cm.