Question

andrea is given $\triangle abc$ and told that $a^2 + b^2 = c^2$, she draws right triangle $rts$ with legs measuring $a$ and $b$ and hypotenuse measuring $x$. which best describes what andrea should do in order to prove that $\triangle abc$ is a right triangle?

Explanation:

Step1: Recall Pythagorean Theorem

For right triangle \( RTS \) with legs \( a, b \) and hypotenuse \( x \), by Pythagorean theorem, \( a^2 + b^2 = x^2 \).

Step2: Compare with Given Equation

Given \( a^2 + b^2 = c^2 \) for \( \triangle ABC \). So \( x^2 = c^2 \), which implies \( x = c \) (since lengths are positive).

Step3: Apply SSS Congruence

Now, \( \triangle ABC \) and \( \triangle RTS \) have sides \( a, b, c \) and \( a, b, x(=c) \) respectively. By SSS (Side - Side - Side) congruence criterion, \( \triangle ABC \cong \triangle RTS \).

Step4: Conclude Right Triangle

Since \( \triangle RTS \) is a right triangle, \( \triangle ABC \) (being congruent to \( \triangle RTS \)) is also a right triangle. So Andrea should show \( x = c \) (by equating \( a^2 + b^2 \) from both triangles) and use SSS congruence to prove \( \triangle ABC \) is right - angled.

Answer:

Andrea should show \( x = c \) (using \( a^2 + b^2 = x^2 \) and \( a^2 + b^2 = c^2 \)) and use SSS congruence to prove \( \triangle ABC\cong\triangle RTS \), hence \( \triangle ABC \) is a right triangle. (The key step is to establish \( x = c \) and use congruence to transfer the right - angle property from \( \triangle RTS \) to \( \triangle ABC \))