August 30th, 2018, 03:20 AM  #1 
Member Joined: Aug 2018 From: Nigeria Posts: 73 Thanks: 2  Need help I'm stuck
A(6,2), B(2,4) and C(8,4) are three points in a plane. P is a point on the line BC and Q is a point in the plane such that PAQB is a rectangle. Find the 1. equations of the sides AP and B 2. coordinates of the Point P 3. area of the rectangle Last edited by skipjack; August 30th, 2018 at 08:31 AM. 
August 30th, 2018, 05:45 AM  #2 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 140 
Where are you stuck? What have you tried?

August 30th, 2018, 06:44 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,636 Thanks: 2620 Math Focus: Mainly analysis and algebra 
I would start with 1) a diagram; and 2) writing the question correctly  B is a point, not a line. Should it be BQ?

August 30th, 2018, 07:17 AM  #4 
Member Joined: Aug 2018 From: Nigeria Posts: 73 Thanks: 2 
equation of AP without given the gradient...Look I'm not asking you to molest me, all I want you to do is to give me hint... I don't need final answer... I just need a responsible person to give me hint.
Last edited by skipjack; August 30th, 2018 at 07:32 AM. 
August 30th, 2018, 08:27 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039 
1. As BC has slope 4/3, AP and BQ have slope 3/4. Hence AP has equation y = (3x  10)/4. I'll let you find the equation of BQ. 2. As BC has equation 4x + 3y = 20, one way to find the coordinates of P is to solve the equations I've given for AP and BC. 3. There are various ways to find the area of the rectangle PAQB. For example, you could work out the lengths of AP and BP and then find their product. The result happens to be an integer. 
August 30th, 2018, 09:10 AM  #6 
Member Joined: Aug 2018 From: Nigeria Posts: 73 Thanks: 2 
Thanks. The slope of BC had been verified, but how did AP and BQ 3/4? Please clarify this.
Last edited by skipjack; August 30th, 2018 at 01:57 PM. 
August 30th, 2018, 01:55 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039 
The product of the slopes of perpendicular lines is 1.

August 30th, 2018, 09:12 PM  #8 
Member Joined: Aug 2018 From: Nigeria Posts: 73 Thanks: 2 
Okay... AP and BQ are each perpendicular to BC.
Last edited by skipjack; August 31st, 2018 at 01:22 AM. 
August 30th, 2018, 09:31 PM  #9 
Member Joined: Aug 2018 From: Nigeria Posts: 73 Thanks: 2 
According to the question, I was asked to find equation of line BP and BQ is 4y  3x = 10 Last edited by skipjack; August 31st, 2018 at 01:23 AM. 
August 31st, 2018, 01:38 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039 
If a line has known slope m/n, its equation can be written as mx  ny = k, where k is found by substituting the coordinates of a known point on the line for x and y. That rule enables you to provide an equation for any line in this question. 

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