Question

17,582 Learners found this answer helpful

are \\( \triangle qrs \\) and \\( \triangle ghi \\) congruent?

Explanation:

Step1: Calculate the side lengths of $\triangle QRS$

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $QR$: $Q(0,-9)$, $R(-9,-9)$.
$$QR=\sqrt{( - 9-0)^2+(-9 + 9)^2}=\sqrt{(-9)^2+0^2}=9$$
For $QS$: $Q(0,-9)$, $S(-1,-1)$.
$$QS=\sqrt{(-1 - 0)^2+(-1+9)^2}=\sqrt{(-1)^2+8^2}=\sqrt{1 + 64}=\sqrt{65}$$
For $RS$: $R(-9,-9)$, $S(-1,-1)$.
$$RS=\sqrt{(-1 + 9)^2+(-1 + 9)^2}=\sqrt{8^2+8^2}=\sqrt{64+64}=\sqrt{128}=8\sqrt{2}$$

Step2: Calculate the side lengths of $\triangle GHI$

For $GH$: $G(1,0)$, $H(1,10)$.
$$GH=\sqrt{(1 - 1)^2+(10 - 0)^2}=\sqrt{0^2+10^2}=10$$
For $GI$: $G(1,0)$, $I(10,0)$.
$$GI=\sqrt{(10 - 1)^2+(0 - 0)^2}=\sqrt{9^2+0^2}=9$$
For $HI$: $H(1,10)$, $I(10,0)$.
$$HI=\sqrt{(10 - 1)^2+(0 - 10)^2}=\sqrt{9^2+(-10)^2}=\sqrt{81 + 100}=\sqrt{181}$$

Step3: Compare the side lengths

Since the side - lengths of $\triangle QRS$ ($9,\sqrt{65},8\sqrt{2}$) and $\triangle GHI$ ($9,10,\sqrt{181}$) are not all equal.

Answer:

No, $\triangle QRS$ and $\triangle GHI$ are not congruent.