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Question 1126749: email: maplespots@gmail.com
Please help me solve this;
"AB has endpoints A(-5,0) and B(4,3). CD has endpoints C(-3,9) and D(1,-3). The equations of the lines containing AB and CD are x - 3y = -5 and 3x + y = 0, respectively."
a. How could you quickly check that these equations are correct?
b. Verify that the lines are perpendicular.
c. Find the point of intersection of AB and CD by solving the system of equations.
d. Find the midpoints of AB and CD. Compare your results with Part c.
e. What kind of quadrilateral is ACBD? Explain your reasoning.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! email: maplespots@gmail.com
Please help me solve this;
"AB has endpoints A(-5,0) and B(4,3). CD has endpoints C(-3,9) and D(1,-3). The equations of the lines containing AB and CD are x - 3y = -5 and 3x + y = 0, respectively."
a. How could you quickly check that these equations are correct?
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Sub the x & y values into the equations.
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b. Verify that the lines are perpendicular.
If the product of the 2 slopes is -1 they're perpendicular.
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c. Find the point of intersection of AB and CD by solving the system of equations.
Do that.
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d. Find the midpoints of AB and CD. Compare your results with Part c.
do that.
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e. What kind of quadrilateral is ACBD? Explain your reasoning.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
Please help me solve this:
"AB has endpoints A(-5,0) and B(4,3). CD has endpoints C(-3,9) and D(1,-3). The equations of the lines
containing AB and CD are x - 3y = -5 and 3x + y = 0, respectively."
a. How could you quickly check that these equations are correct?
The simplest way to check that AB lies on the straight line x - 3y = -5 is to check that each endpoint A and B lies on this line.
To check it for AB, simply substitute the coordinates of A x= -5 and y= 0 into the equation:
-5 -3*0 = - 5 (! correct !),
and then substitute the coordinates of B x= 4 and y= 3 into the equation:
4 -3*3 = - 5 (! correct !).
Thus AB is checked. Now you check for CD by the same way.
b. Verify that the lines are perpendicular.
The line x - 3y = -5 has the slope  .
The line 3x + y = 0 has the slope -3.
The numbers and -3 are opposite reciprocal.
It means that the two lines are perpendicular.
c. Find the point of intersection of AB and CD by solving the system of equations.
x - 3y = -5. (1)
3x + y = 0. (2)
From eq(1) express x = -5 + 3y and substitute it into eq(2). You will get
3*(-5+3y) + y = 0,
-15 + 9y + y = 0
10y = 15 ====> y = 15/10 = 1.5.
Then x = -5 + 3y = -5 +3*1.5 = -5 + 4.5 = -0.5.
Thus the intersection point is (x,y) = (-0.5,1.5)
d. Find the midpoints of AB and CD. Compare your results with Part c.
To find midpoint of AB, take x-coordinates of A and B, add them and divide the sum by 2
(so you will get the average of x-coordinates of endpoints A and B).
By doing it, you will get x-coordinate of the midpoint AB as  = -0.5.
Now make the same with y-coordinates of A and B: you will get y-coordinate of the midpoint AB as  = 1.5.
Thus the midpoint of AB is (x,y) = (-0.5,1.5).
To determine the midpoint of CD, do the same with the points C and D.
Notice that is is THE SAME point as the intersection point of AB and CD, which we determined in Part c).
e. What kind of quadrilateral is ACBD? Explain your reasoning.
The quadrilateral ABCD has perpendicular diagonals AB and CD; and the diagonals intersetion is the midpoint of each diagonal.
Hence the quadrilateral ABCD is a rhombus.
COMPLETED.
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