Question

given the graph, find mn. round to the nearest hundredth. distance formula
if $overline{bd}$ bisects $angle abc$, $mangle dbc = 79^{circ}$, and $mangle abc=(9x - 4)^{circ}$, find the value of $x$.

Explanation:

Step 1: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. First, assume the coordinates of point $M$ are $(x_1,y_1)$ and of point $N$ are $(x_2,y_2)$ from the graph. Let's say $M(- 2,2)$ and $N(2,0)$.

Step 2: Substitute the values into the formula

$d=\sqrt{(2-(-2))^2+(0 - 2)^2}=\sqrt{(2 + 2)^2+(0 - 2)^2}=\sqrt{4^2+( - 2)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\approx4.47$.

Step 3: Solve the angle - bisector problem

Since $\overline{BD}$ bisects $\angle ABC$, then $m\angle ABC=2m\angle DBC$. Given $m\angle DBC = 79^{\circ}$, so $m\angle ABC=2\times79^{\circ}=158^{\circ}$.
Also, given $m\angle ABC=(9x - 4)^{\circ}$. Then we set up the equation $9x-4 = 158$.

Step 4: Solve the equation for $x$

Add 4 to both sides of the equation: $9x-4 + 4=158 + 4$, which gives $9x=162$. Divide both sides by 9: $x=\frac{162}{9}=18$.

Answer:

The length of $MN\approx4.47$ and $x = 18$.