Question

big idea questions
creating and carrying out a plan
one of the issues we all have with approaching complex application questions is we sometimes forget to map out what we are intending to do before beginning the problem. for the following question, plan out each of the steps you would carry out on the path to the solution in words. once you have completed that, execute those steps to determine the solution to the problem.

• in the graph provided
• (overline{mn}perpoverline{ba})
• (overline{ab}perpoverline{bc})
• point (a) divides segment (overline{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 (overline{bc}) to the nearest hundredth of a unit?
numbered steps in words:
carrying out those steps:

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, \(m = 2\), \(n = 1\), \(x_1=120\), \(y_1 = 480\), \(x_2=160\), \(y_2=310\).

$$x=\frac{2\times160+1\times120}{2 + 1}=\frac{320 + 120}{3}=\frac{440}{3}\approx146.67$$

$$y=\frac{2\times310+1\times480}{2 + 1}=\frac{620+480}{3}=\frac{1100}{3}\approx366.67$$

So, the coordinates of \(A\) are \((\frac{440}{3},\frac{1100}{3})\)

Step2: Find the slope of \(MN\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(M(120,480)\) and \(N(160,310)\), \(m_{MN}=\frac{310 - 480}{160 - 120}=\frac{- 170}{40}=-\frac{17}{4}\)
Since \(MN\perp BA\), the slope of \(BA\) (\(m_{BA}\)) is the negative reciprocal of \(m_{MN}\). So, \(m_{BA}=\frac{4}{17}\)

Step3: Find the equation of line \(BA\)

The equation of a line in point - slope form is \(y - y_0=m(x - x_0)\), where \((x_0,y_0)\) is a point on the line. Using point \(A(\frac{440}{3},\frac{1100}{3})\) and \(m=\frac{4}{17}\), \(y-\frac{1100}{3}=\frac{4}{17}(x - \frac{440}{3})\)
Since \(B\) is on the \(y\) - axis (\(x = 0\)), substitute \(x = 0\) into the equation:

$$y-\frac{1100}{3}=\frac{4}{17}(0-\frac{440}{3})$$

$$y=\frac{1100}{3}-\frac{1760}{51}=\frac{18700 - 1760}{51}=\frac{16940}{51}\approx332.16$$

So, \(B=(0,\frac{16940}{51})\)

Step4: Find the equation of line \(AB\) and then use the perpendicular relationship to find the equation of line \(BC\)

Since \(AB\perp BC\) and \(m_{AB}=\frac{4}{17}\), \(m_{BC}=-\frac{17}{4}\)
The equation of line \(BC\) using point - slope form with point \(B(0,\frac{16940}{51})\) is \(y-\frac{16940}{51}=-\frac{17}{4}(x - 0)\)
When \(y = 0\) (since \(C\) is on the \(x\) - axis), \(0-\frac{16940}{51}=-\frac{17}{4}x\)

$$x=\frac{16940\times4}{51\times17}=\frac{67760}{867}\approx78.15$$

Answer:

The length of \(BC\) is approximately \(78.15\) units.