20. The diagram shows a trapezium in which AD is parallel to BC and angle ADC = angle BCD = 90°. The points A, B and C are (a, 18), (12, -2) and (2, -7) respectively. Given that AB 2BC, find (a) the value of a, (b) the equation of AD, (c) the equation of CD, (d) the coordinates of D, (e) the area of the trapezium.

Answer: (a) Given AB is twice BC and the coordinates of A, B, and C:

A(a,18), B(12,-2), C(2,-7)

Let's find the distances between B and C and between A and B.

Distance BC = √[(2 - 12)² + (-7 - (-2))²] = √((-10)² + (-5)²) = √(100 + 25) = √125 = 5√5

The distance AB is twice the distance BC:

AB = 2BC = 2 * 5√5 = 10√5

Now find the length AB using A and B's coordinates:

AB = √[(a - 12)² + (18 - (-2))²] = √[(a - 12)² + 20²] = √[(a - 12)² + 400]

We can equate this with 10√5:

√[(a - 12)² + 400] = 10√5 (a - 12)² + 400 = (10√5)² (a - 12)² + 400 = 500

We can now solve for a:

(a - 12)² = 500 - 400 (a - 12)² = 100 a - 12 = 10 or a - 12 = -10

This gives us two values for a:

a1 = 12 + 10 = 22 a2 = 12 - 10 = 2

Since A and B are different points and B has an x-value of 12, we discard the solution a2 = 2. Therefore, the value of a is:

a = 22

(b) To find the equation of AD, we need to know the slope of AD. Since AD is parallel to BC, the slope of AD is the same as the slope of BC. However, we know that ADC is a right angle; hence, AD is a vertical line, and its equation is simply the x-coordinate of A, which is:

x = 22

(c) Again, the equation of line CD would be horizontal as CDC is a right angle; hence, it has a slope of 0. The y-coordinate of C gives us the equation of CD:

y = -7

(d) The coordinates of D would have the same x as A and the same y as C:

D = (22, -7)

(e) The area of the trapezium can be found using the formula for the area of a trapezoid:

Area = 0.5 * (sum of parallel sides) * height

We have AD and BC as parallel sides, so we need to calculate their lengths:

AD = A's y-coordinate - D's y-coordinate = 18 - (-7) = 25

BC is already calculated as 5√5. Therefore:

Area = 0.5 * (AD + BC) * DC = 0.5 * (25 + 5√5) * 10 = 0.5 * (25 + 5√5) * (12 - 2) = 0.5 * (25 + 5√5) * 10 = 0.5 * (25 + 5√5) * 10 = (12.5 + 2.5√5) * 10 = 125 + 25√5 square units.