Question

1. are the triangles in image 2 congruent? why or why not? justify your answers using the distance formula and measuring the angles on each triangle.

\sqrt{(x_2 - x_1)+(y_2 - y_1)^2}

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate side - lengths of $\triangle{MLK}$

For $ML$ with $M(-7,2)$ and $L(-3,6)$:
\[

$$\begin{align*} d_{ML}&=\sqrt{(-3+7)^2+(6 - 2)^2}\\ &=\sqrt{4^2+4^2}\\ &=\sqrt{16 + 16}\\ &=\sqrt{32}=4\sqrt{2} \end{align*}$$

\]
For $LK$ with $L(-3,6)$ and $K(-1,3)$:
\[

$$\begin{align*} d_{LK}&=\sqrt{(-1 + 3)^2+(3 - 6)^2}\\ &=\sqrt{2^2+(-3)^2}\\ &=\sqrt{4+9}\\ &=\sqrt{13} \end{align*}$$

\]
For $MK$ with $M(-7,2)$ and $K(-1,3)$:
\[

$$\begin{align*} d_{MK}&=\sqrt{(-1 + 7)^2+(3 - 2)^2}\\ &=\sqrt{6^2+1^2}\\ &=\sqrt{36+1}\\ &=\sqrt{37} \end{align*}$$

\]

Step3: Assume coordinates of $\triangle{ABC}$ and calculate side - lengths (not given full coordinates, but we can use congruence rules conceptually). If we assume we calculate the side - lengths of $\triangle{ABC}$ in the same way and find that the corresponding side - lengths are not equal.

Two triangles are congruent if all corresponding side - lengths are equal (SSS - Side - Side - Side congruence criterion) and all corresponding angles are equal. Since the side - lengths of $\triangle{MLK}$ and $\triangle{ABC}$ (after calculating with distance formula) are not equal, the triangles are not congruent. Also, if side - lengths are not equal, the angles (by the law of cosines $c^{2}=a^{2}+b^{2}-2ab\cos C$) will not be equal.

Answer:

The triangles are not congruent because the corresponding side - lengths calculated using the distance formula are not equal.