8. (a) Find an equation of the line joining A(7, 4) and B(2, 0), giving your answer in the form ax+by+c=0, where a, b and c are integers - Edexcel - A-Level Maths Pure - Question 10 - 2010 - Paper 1

To compute the length of segment AB, we use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substituting the coordinates of A(7, 4) and B(2, 0):

$AB = \sqrt{(2 - 7)^2 + (0 - 4)^2} = \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41}$

Therefore, the length of AB is: $\sqrt{41}$

Given that AC = AB and the coordinates of point C are (2, t), we apply the distance formula:

$AC = \sqrt{(2 - 7)^2 + (t - 4)^2} = \sqrt{(-5)^2 + (t - 4)^2} = \sqrt{25 + (t - 4)^2}$

Setting AC equal to AB:

$\sqrt{25 + (t - 4)^2} = \sqrt{41}$

Squaring both sides:

$25 + (t - 4)^2 = 41$

Solving for (t - 4)^2 yields:

$(t - 4)^2 = 41 - 25 = 16$

Taking the square root gives:

t - 4 = 4 or t - 4 = -4, leading to:

t = 8 ext{ or } t = 0.

Since t must be greater than 0, we find:

t = 8.

The area (A) of a triangle formed by three points can be calculated using the formula:

$A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

For triangle ABC with A(7, 4), B(2, 0), and C(2, 8):

Substituting the coordinates:

$A = \frac{1}{2} |7(0 - 8) + 2(8 - 4) + 2(4 - 0)|$

Calculating each term:

$= \frac{1}{2} |7(-8) + 2(4) + 2(4)|$ $= \frac{1}{2} |-56 + 8 + 8|$ $= \frac{1}{2} |-40| = 20$

Therefore, the area of triangle ABC is: $20$.