Question

the vertices of δghj are g(-2, -3), h(-2,5), and j(4,5). if δklm≅δghj, find kl, lm, and km. kl = (type an integer or a decimal.)

Explanation:

Answer:

First, find the lengths of the sides of \(\triangle GHJ\) using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(GH\) with \(G(-2,-3)\) and \(H(-2,5)\):
\[

$$\begin{align*} d_{GH}&=\sqrt{(-2+ 2)^2+(5 + 3)^2}\\ &=\sqrt{0+64}\\ &=8 \end{align*}$$

\]
For \(HJ\) with \(H(-2,5)\) and \(J(4,5)\):
\[

$$\begin{align*} d_{HJ}&=\sqrt{(4 + 2)^2+(5 - 5)^2}\\ &=\sqrt{36+0}\\ &=6 \end{align*}$$

\]
For \(GJ\) with \(G(-2,-3)\) and \(J(4,5)\):
\[

$$\begin{align*} d_{GJ}&=\sqrt{(4 + 2)^2+(5 + 3)^2}\\ &=\sqrt{36 + 64}\\ &=\sqrt{100}\\ &=10 \end{align*}$$

\]
Since \(\triangle KLM\cong\triangle GHJ\), corresponding - sides are equal.
If \(KL\) corresponds to \(GH\), then \(KL = 8\).
If \(LM\) corresponds to \(HJ\), then \(LM = 6\).
If \(KM\) corresponds to \(GJ\), then \(KM = 10\).

So:
KL = 8
LM = 6
KM = 10