Question

• in the graph provided
• \\( \overline { m n } \perp \overline { b c } \\)
• \\( \overline { a b } \perp \overline { b c } \\)
• point a divides

segment \\( \overline { m n } \\) 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 \\( \overline { b c } \\)
to the nearest hundredth of a unit?

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, \(m = 2\), \(n = 1\), \(x_1=120\), \(y_1 = 430\), \(x_2=180\), \(y_2 = 310\).

$$ x=\frac{2\times180+1\times120}{2 + 1}=\frac{360+120}{3}=\frac{480}{3}=160 $$

$$ y=\frac{2\times310+1\times430}{2 + 1}=\frac{620 + 430}{3}=\frac{1050}{3}=350 $$

So, \(A=(160,350)\)

Step2: Since \(AB\perp BC\) and \(B\) is on the \(y\) - axis (\(x = 0\)) and \(A=(160,350)\)

The slope of \(AB\) is undefined (because \(AB\perp BC\) and \(BC\) is horizontal or vertical. Since \(B\) is on \(y\) - axis and \(A\) has \(x = 160\), the line \(AB\) is vertical). So, the \(x\) - coordinate of \(A\) and \(B\) for the vertical line \(AB\): \(B=(0,350)\)

Step3: Calculate the length of \(BC\)

Since \(C\) is on the \(x\) - axis (\(y = 0\)) and \(B=(0,350)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, for \(B(0,350)\) and \(C\) (let \(C=(x,0)\), but since \(BC\) is a vertical or horizontal line. As \(AB\perp BC\) and \(AB\) is vertical, \(BC\) is horizontal. The \(y\) - coordinate of \(B\) is \(350\) and \(C\) is on \(x\) - axis (\(y = 0\)), and using the fact that the \(x\) - coordinate of \(B\) is \(0\). The length of \(BC\) is the difference in \(y\) - coordinates (because \(AB\) is vertical and \(BC\) is horizontal).

$$ BC=\sqrt{(0 - 0)^2+(350-0)^2}=350.00 $$

Answer:

\(350.00\)