Please help me find the solution to this question! Thank you!

Question

The y-interepts, P and Q, of two perpendicular lines intersecting at the point A(6,8)have a sum of zero. What is the area of ΔAPQ?

Answer 1

Hi Caroline,

 

Sounds like a challenging problem for sure, but we'll step through it and all will be clear.

 

Let's begin by determining what are the base and the altitude of triangle APQ, since these are the two items needed for calculating the area of a triangle:

    1.  The base will be side PQ since we are told that points P and Q both lie on the y-axis (these points being y-intercepts).  We are further told that the sum of these y-intercepts is zero.  This means that each y-intercept is the additive inverse of the other and can be represented as "b" and "-b", respectively. Therefore, the length of the base of the triangle will be the difference of these two y-intercepts (b - (-b)) or 2b units.

    2.  The altitude of the triangle will be the horizontal distance from point A to the portion of the y-axis bounded by points P and Q (the vertical base of the triangle).  This distance is the difference between the x coordinate of point A (6)  and the x coordinate of the y-axis (0):  6 - 0 (6 units).

 

All that is needed now is to determine the values of our two y-intercepts (b and -b).  

 

Let's begin by putting the equations of the lines representing sides AP and AQ into slope-intercept form (noting that we are told these lines are perpendicular to each other and, therefore, their slopes are the negative reciprocals of each other):

    1.  Line AP will descend slightly from point P (0,b) to point A (6,8), so it will have a small negative slope and a positive y-intercept:

            y = -(1/m)x + b

    2.  Line AQ will ascend steeply from point Q (0,-b) to point A (6,8), so it will large positive slope and a negative y-intercept:

            y = mx - b

 

Now let's plug in the coordinates point A (common intersection point of these two lines) into the slope-intercept equations for both lines and then add the equations together to eliminate b and solve for m:

           8 = -(1/m)(6) + b  [equation of line AP] 

           8 = m(6) - b         [equation for line AQ]

          16 = 6m - (6/m)     [equations added together]

          (m)(16) = (m)(6m) - m(6/m)

          16m = 6m² -6

          6m² - 16m - 6 = 0

          2(3m² - 8m - 3) = 0

          3m² - 8m - 3 = 0

          3m² + (-9 + 1)m - 3 =0

          3m² - 9m + m - 3 = 0

          3m(m - 3) + 1(m - 3) = 0

          (m - 3)(3m + 1) = 0

          m - 3 = 0

               m = 3

          3m + 1 = 0

          3m = -1

            m = -1/3

 

We can now see that the roots of  the quadratic equation formed from the addition of our two line equations, gives us the two negative reciprocal slopes (-1/3 and 3) of our two perpendicular lines forming sides AP and AQ, respectively.

 

We are now in position to calculate the absolute value of y-intercept (b) by plugging into either of our two line equations:  our slope and the coordinates of common point A:

    8 = (6)(-1/3) + b  [equation of line AP] 

    8 = -2 + b

    b = 8 + 2

    b = 10 (absolute value)

    8 = (3)(6) - b       [equation of line AQ]

    8 = 18 - b

    b = 18 - 8

    b = 10 (absolute value)

 

So now we know that since the absolute value of our y-intercept (b) is 10, we also know that our positive and negative y-intercepts are 10 and -10, respectively. Therefore, the length of base P is

(10 - (-10)) = 10 + 10 = 20

 

Putting it all together:

   Area of triangle APQ = 1/2(base x height)

   Area of triangle APQ = 1/2(PQ)(altitude)

   Area of triangle APQ = 1/2(20)(6)

   Area of triangle APQ = 1/2(120)

   Area of triangle APQ = 60 square units

 

Below is the link to our diagram for this problem showing the vertices of triangle APQ with altitude AB drawn to base PQ, so you can have a visual aid supporting our solution:

 

https://dl.dropbox.com/s/uvobu3n2617a5nz/Triangle.png?raw=1

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.