in the graph provided
·mn ⊥ bc
·ab ⊥ bc
·point a divides segment mn in a 2:1 ratio from point m
·point b is on the y - axis
·point c is on the x - axis
what is the length of bc to the nearest hundredth of a unit?

Question

Accepted Answer

# Explanation:
## Step1: Find the coordinates of point $A$
Use the section formula. If a point $A(x,y)$ divides the line - segment joining $M(x_1,y_1)$ and $N(x_2,y_2)$ in the ratio $m:n$ from $M$, then $x=\frac{mx_2 + nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 2$, $n = 1$, $x_1=120$, $y_1 = 430$, $x_2=180$, $y_2=310$.
$$
\begin{align*}
x&=\frac{2\times180+1\times120}{2 + 1}=\frac{360+120}{3}=\frac{480}{3}=160\
y&=\frac{2\times310 + 1\times430}{2+1}=\frac{620 + 430}{3}=\frac{1050}{3}=350
\end{align*}
$$
So, the coordinates of $A$ are $(160,350)$.

## Step2: Since $B$ is on the $y$ - axis ($x = 0$) and $C$ is on the $x$ - axis ($y = 0$), and $AB\perp BC$ and $MN\perp BC$. The slope of $MN$ is $m_{MN}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{310-430}{180 - 120}=\frac{-120}{60}=-2$. Since $MN\perp BC$, if the slope of $MN$ is $m_{MN}$ and the slope of $BC$ is $m_{BC}$, then $m_{MN}\times m_{BC}=-1$. So, $m_{BC}=\frac{1}{2}$.
Let $B=(0,b)$ and $C=(c,0)$. The slope of $BC$ is $m_{BC}=\frac{0 - b}{c-0}=\frac{-b}{c}=\frac{1}{2}$, so $c=-2b$.
The vector $\overrightarrow{AB}=(0 - 160,b - 350)=(-160,b - 350)$ and $\overrightarrow{BC}=(c-0,0 - b)=(c,-b)$. Since $AB\perp BC$, $\overrightarrow{AB}\cdot\overrightarrow{BC}=0$. Substitute $c=-2b$ into $\overrightarrow{AB}\cdot\overrightarrow{BC}$:
$$
\begin{align*}
(-160)\times(-2b)+(b - 350)\times(-b)&=0\
320b-b^{2}+350b&=0\
b^{2}-670b&=0\
b(b - 670)&=0
\end{align*}
$$
$b = 0$ is not valid (as it will make the points degenerate). So, $b = 670$, then $c=-2\times670=-1340$.

## Step3: Calculate the length of $BC$
Use the distance formula $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$. For $B=(0,670)$ and $C=(-1340,0)$
$$
\begin{align*}
BC&=\sqrt{(0+1340)^{2}+(670 - 0)^{2}}\
&=\sqrt{1340^{2}+670^{2}}\
&=\sqrt{670^{2}(4 + 1)}\
&=670\sqrt{5}\
&\approx670\times2.236\
&=1508.12
\end{align*}
$$

# Answer:
$1508.12$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the coordinates of point \(A\)

Step3: Calculate the length of \(BC\)

Answer: