# Suppose we draw semicircles on the sides of a triangle and compute their areas. If the sum of the areas of two of the semicircles equals the area of the third, is the triangle a right triangle?

Yes.
(pi/2)(a^2 + b^2) = (pi/2) c^2
if and only if
a^2 + b^2 = c^2,
and that requires a right angle triangle.

## To understand why the triangle must be a right triangle in this scenario, let's break down the steps.

We start by drawing semicircles on the sides of a triangle. Let's call the sides of the triangle a, b, and c, with c being the longest side.

The area of a semicircle is given by (π/2) * r^2, where r is the radius of the semicircle.

Let's say the areas of two semicircles on sides a and b are Aa and Ab, respectively, and the area of the semicircle on side c is Ac.

Given that the sum of the areas of two semicircles equals the area of the third, we have:

Aa + Ab = Ac

Substituting the formula for the area of a semicircle, we get:

(π/2) * ra^2 + (π/2) * rb^2 = (π/2) * rc^2

Canceling out (π/2) from both sides, we get:

ra^2 + rb^2 = rc^2

Comparing this equation with the Pythagorean theorem a^2 + b^2 = c^2, we observe that they have the same form.

This means that if the sum of the areas of two semicircles equals the area of the third, then the equation ra^2 + rb^2 = rc^2 is satisfied. This equation is only true for right triangles, where the Pythagorean theorem holds.

Therefore, if the sum of the areas of two semicircles on the sides of a triangle equals the area of the third semicircle, then the triangle must be a right triangle.
The text contains a mathematical question about the relationship between the areas of semicircles drawn on the sides of a triangle and whether this implies the triangle is a right triangle. The text also provides a detailed answer to this question, including a step-by-step explanation and the application of the Pythagorean theorem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Suppose we draw semicircles on the sides of a triangle and compute their areas. If the sum of the areas of two of the semicircles equals the area of the third, is the triangle a right triangle?

Yes.
(pi/2)(a^2 + b^2) = (pi/2) c^2
if and only if
a^2 + b^2 = c^2,
and that requires a right angle triangle.

## To understand why the triangle must be a right triangle in this scenario, let's break down the steps.

We start by drawing semicircles on the sides of a triangle. Let's call the sides of the triangle a, b, and c, with c being the longest side.

The area of a semicircle is given by (π/2) * r^2, where r is the radius of the semicircle.

Let's say the areas of two semicircles on sides a and b are Aa and Ab, respectively, and the area of the semicircle on side c is Ac.

Given that the sum of the areas of two semicircles equals the area of the third, we have:

Aa + Ab = Ac

Substituting the formula for the area of a semicircle, we get:

(π/2) * ra^2 + (π/2) * rb^2 = (π/2) * rc^2

Canceling out (π/2) from both sides, we get:

ra^2 + rb^2 = rc^2

Comparing this equation with the Pythagorean theorem a^2 + b^2 = c^2, we observe that they have the same form.

This means that if the sum of the areas of two semicircles equals the area of the third, then the equation ra^2 + rb^2 = rc^2 is satisfied. This equation is only true for right triangles, where the Pythagorean theorem holds.

Therefore, if the sum of the areas of two semicircles on the sides of a triangle equals the area of the third semicircle, then the triangle must be a right triangle.
user
The text contains a mathematical question about the relationship between the areas of semicircles drawn on the sides of a triangle and whether this implies the triangle is a right triangle. The text also provides a detailed answer to this question, including a step-by-step explanation and the application of the Pythagorean theorem.

TYPE: SINGLE_QUESTION_WITH_ANSWER