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 $\triangle abc$ a right triangle?
show that $c = x$, so $\triangle abc \sim \triangle rts$ and so $\angle c \sim \angle s$. this means $\angle c$ is a right angle, and so $\triangle abc$ is a right triangle.
show that $c = x$, so $\triangle abc \cong \triangle rts$ and so $\angle c \cong \angle s$. this means $\angle c$ is a right angle, and so $\triangle abc$ is a right triangle.
show that $a^2 + b^2 = c^2$, so $\angle a \cong \angle r$. this means $\triangle abc \cong \triangle rts$, and so $\angle c$ is a right angle

Explanation:

Step1: Recall Pythagorean Theorem

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

Step2: Analyze Triangle Congruence

\( \triangle ABC \) has sides \( a, b, c \); \( \triangle RTS \) has sides \( a, b, x \). Since \( c = x \), by SSS (Side - Side - Side) congruence criterion, \( \triangle ABC \cong \triangle RTS \).

Step3: Determine Right Angle

\( \triangle RTS \) is a right triangle with right angle \( \angle S \). Since \( \triangle ABC \cong \triangle RTS \), corresponding angles are equal, so \( \angle C\cong\angle S \), meaning \( \angle C \) is a right angle, so \( \triangle ABC \) is a right triangle. The second option correctly uses SSS congruence (show \( c = x \) for SSS, leading to \( \triangle ABC\cong\triangle RTS \) and \( \angle C\cong\angle S \) (right angle)).

Answer:

Show that \( c = x \), so \( \triangle ABC\cong\triangle RTS \) and so \( \angle C\cong\angle S \). This means \( \angle C \) is a right angle, and so \( \triangle ABC \) is a right triangle.