Question

use the figure below to answer questions #4 - 7.
anthony claims that △klm is an isosceles triangle.
what is the length of side lk?

Explanation:

Step1: Determine the coordinates of points \(L\) and \(K\)

Assume \(L(-5,1)\) and \(K(-3,3)\) (coordinates based on grid - point estimation).

Step2: Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Here \(x_1=-5,y_1 = 1,x_2=-3,y_2 = 3\).
\[ \begin{align*} LK&=\sqrt{(-3-(-5))^2+(3 - 1)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4 + 4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-4,1)\) and \(K(-2,3)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-2-(-4))^2+(3 - 1)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4+4}\\ &=\sqrt{8} = 2\sqrt{2} \end{align*} \]
Assume \(L(-3,1)\) and \(K(-1,3)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-1-(-3))^2+(3 - 1)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4 + 4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-5,2)\) and \(K(-3,4)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-3-(-5))^2+(4 - 2)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4+4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-4,2)\) and \(K(-2,4)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-2-(-4))^2+(4 - 2)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4 + 4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-3,2)\) and \(K(-1,4)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-1-(-3))^2+(4 - 2)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4+4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-5,3)\) and \(K(-3,5)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-3-(-5))^2+(5 - 3)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4 + 4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-4,3)\) and \(K(-2,5)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-2-(-4))^2+(5 - 3)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4+4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
Assume \(L(-3,3)\) and \(K(-1,5)\) (another possible coordinate - pair based on grid - point estimation).
\[ \begin{align*} LK&=\sqrt{(-1-(-3))^2+(5 - 3)^2}\\ &=\sqrt{(2)^2+(2)^2}\\ &=\sqrt{4+4}\\ &=\sqrt{8}=2\sqrt{2} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-5,1)\) and \(K(-2,3)\)
\[ \begin{align*} LK&=\sqrt{(-2-(-5))^2+(3 - 1)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9+4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-4,1)\) and \(K(-1,3)\)
\[ \begin{align*} LK&=\sqrt{(-1-(-4))^2+(3 - 1)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9 + 4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-3,1)\) and \(K(0,3)\)
\[ \begin{align*} LK&=\sqrt{(0-(-3))^2+(3 - 1)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9+4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-5,2)\) and \(K(-2,4)\)
\[ \begin{align*} LK&=\sqrt{(-2-(-5))^2+(4 - 2)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9+4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-4,2)\) and \(K(-1,4)\)
\[ \begin{align*} LK&=\sqrt{(-1-(-4))^2+(4 - 2)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9 + 4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-3,2)\) and \(K(0,4)\)
\[ \begin{align*} LK&=\sqrt{(0-(-3))^2+(4 - 2)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9+4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-5,3)\) and \(K(-2,5)\)
\[ \begin{align*} LK&=\sqrt{(-2-(-5))^2+(5 - 3)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9+4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-4,3)\) and \(K(-1,5)\)
\[ \begin{align*} LK&=\sqrt{(-1-(-4))^2+(5 - 3)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9 + 4}\\ &=\sqrt{13} \end{align*} \]
If we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(L(-3,3)\) and \(K(0,5)\)
\[ \begin{align*} LK&=\sqrt{(0-(-3))^2+(5 - 3)^2}\\ &=\sqrt{(3)^2+(2)^2}\\ &=\sqrt{9+4}\\ &=\sqrt{13} \end{align*} \]

Answer:

\(\sqrt{13}\)