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In the xy-plane, a trapezium ABCD has one of its parallel sides AB on the x-axis with vertex A at the origin. The x-coordinate of point B is 6 and the length of the smaller parallel side CD is 2 less than the length of the longer parallel side. If the side AD lies on the line with the equation y = x and the area of the trapezium is 5 square units, what is the coordinate of point C?
  • a)
    (1,1)
  • b)
    (1,5)
  • c)
    (5,1)
  • d)
    (5,5)
  • e)
    (6,1)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In the xy-plane, a trapezium ABCD has one of its parallel sides AB on ...
Given
Trapezium ABCD with AB on x-axis
  • Coordinates of A = (0,0)
  • x-coordinate of B =6
    • Coordinate of B (6,0)
  • CD = AB – 2
  • AD lies on the line y =x
    • Let’s assume the coordinate of point D (a, a)
  • Ar(ABCD) = 5
To Find: Coordinates of point C?
Approach:
  • As AB lies on x-axis and CD is parallel to AB, the y-coordinate of point C will be equal to y-coordinate of point D, i.e. a
  • The x-coordinate of point C will be x-coordinate of point D + length of CD, i.e. (a + CD)
    • Now, we have expressed the coordinates of point C in terms of a and length of CD
  • Now, we are given that the length of CD is 2 less than the length of AB. As we know the coordinates of points A and B, we can find the length of AB and hence the length of CD.
  • So, we now need to find the value of a to find the coordinates of point C We know that area of trapezium ABCD = ½ * (AB + CD) * distance between AB and CD
    • As we are given the value of the area of the trapezium, AB and CD, we will get the distance between the parallel lines from the above equation.
    • Now, the distance between the parallel lines will be equal to the y-coordinate of point C. Thus, we will get the value of a from here.
Working out
  • Length of AB = x-coordinate of point B – x coordinate of point A = 6 – 0 = 6 units
  • Hence, length of CD = 6 – 2 = 4 units
  • So, coordinate of point C = (a + CD, a) = (a+4, a)
  • Area of trapezium ABCD = ½ *(AB+CD) * distance between AB and CD
    • As distance between AB and CD is equal to the y-coordinate of point C, we can write
    • Area if trapezium = 5 = ½ *(6+4) * (a)
    • Thus a = 1 and a+4 = 5
    • Hence, coordinate of point C (5, 1)
  • Thus, the correct answer is Option C
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In the xy-plane, a trapezium ABCD has one of its parallel sides AB on the x-axis with vertex A at the origin. The x-coordinate of point B is 6 and the length of the smaller parallel side CD is 2 less than the length of the longer parallel side. If the side AD lies on the line with the equation y = x and the area of the trapezium is 5 square units, what is the coordinate of point C?a)(1,1)b)(1,5)c)(5,1)d)(5,5)e)(6,1)Correct answer is option 'C'. Can you explain this answer?
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