a We are given three points A(6,8), O(0,0), and B(10,0).
We are asked to write equations of lines l and m such that l ⊥ OA at A, and m ⊥ OB at B.
Equation of l
We will write the equation of line l using the slope-intercept form, which is as follows.
y=mx+b
In the above equation m is the slope and b the y-intercept of the line. To find the slope m, we will use the fact that l is perpendicular to OA. This means that the product of their slopes is - 1. Let's find the slope of OA.
We can see above that the slope of OA is 86. Therefore, we have the following.
m* 8/6=- 1
Let's solve the above equation to find the value of m.
We found that the slope of l is m=- 34 and thus we can partially write its equation.
y=- 3/4x+b
Since we are told A(6,8) is on l, to find the y-intercept b we will substitute 6 for x and 8 for y in the above equation.
We found that the y-intercept b is 252 and thus the equation of line l is as follows.
Line l: y=- 3/4x+25/2
Let's draw the lines OA and l to see how they look.
Equation of m
To write the equation of line m, we will use the fact that m ⊥ OB. Let's consider the line OB.
We can see above that OB is a horizontal line. Therefore, m will be a vertical line passing through B(10,0) and thus its equation is as follows.
Line m: x=10
Let's draw line m and OB to see how they look.
b We are asked to find the coordinates of C, the point of intersection of lines l and m. To do so, we will solve the System of Linear Equations formed by the equations of the lines.
y=- 34x+ 252 & (I) x=10 & (II)
From Equation (II) we know that x=10. To find the value of y, we will substitute 10 for x in Equation (I), and solve for y.
We have found that the point of intersection of the lines is C(10,5). Let's graph both lines to check our answer.
We can see above that the point of intersection of the lines is C(10,5) and thus our answer is correct.
c We are asked to show that CA=CB. To do so, we will use the Distance Formula to find CA and CB.
d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)To find CA, the distance between C(10,5) and A(6,8), we will substitute (10,5) for (x_1,y_1) and (6,8) for (x_2,y_2) in the above formula.
We have found that CA=5. Let's now find CB, the distance between C(10,5) and B(10,0), by substituting (10,5) and (10,0) for (x_1,y_1) and (x_2,y_2) in the Distance Formula.
