Chapter 8.2, Problem 50HP

To determine

To explain:

Given statement is true or false.

Answer to Problem 50HP

Any two right triangles with the same hypotenuse have the same area is false.

Explanation of Solution

Given information:

Statement:

Any two right triangles with the same hypotenuse have the same area.

In order to find a counterexample of the given statement, one should keep the hypothesis true, and falsify the conclusion. That is find two right triangles with the same hypotenuse and with different areas.

So, consider the following two right triangles $△BCD$ and $△BEC$ .

  

The right $△BCD$ with hypotenuse $2$ and legs $\sqrt{2}$ and $\sqrt{2}$

Pythagoras theorem is satisfied because, ${\left(\sqrt{2}\right)}^2+{\left(\sqrt{2}\right)}^2=2^2$

The area of $△BCD$ is $\frac{1}{2}\left(\sqrt{2}\right)\left(\sqrt{2}\right)=1$

The right $△BEC$ is with hypotenuse $2$ and legs $1$ and $\sqrt{3}$

Pythagorean Theorem is satisfied, because $1^2+{\left(\sqrt{3}\right)}^2=2^2$

The area of $△BEC$ is $\frac{1}{2}\left(1\right)\left(\sqrt{3}\right)=\frac{\sqrt{3}}{2}$

Notice that, $1\ne \frac{\sqrt{3}}{2}$ . This is counterexample to the given statement, because the hypothesis is true, and the conclusion is false.

Hence, the given statement is false.