The points A, B, and C have co-ordinates as follows: A (3, 5) B (-6, 2) C (-4, -4) (a) Plot A, B, and C on the diagram - Leaving Cert Mathematics - Question 4 - 2012

To find the equation of line AB, we first need the slope (m) and the y-intercept (b).

1. Calculate the slope using the formula:

   $m = \frac{y_2 - y_1}{x_2 - x_1}$

   Where:

   • A (3, 5) is (x1, y1)
   • B (-6, 2) is (x2, y2)

   Plugging in the values:

   $m = \frac{2 - 5}{-6 - 3} = \frac{-3}{-9} = \frac{1}{3}$

2. Now, use point A to find the y-intercept. The line equation in point-slope form is:

   $y - y_1 = m(x - x_1)$

   So:

   $y - 5 = \frac{1}{3}(x - 3)$

   Expanding this gives:

   $y - 5 = \frac{1}{3}x - 1$

   Thus, the equation of the line AB is:

   $y = \frac{1}{3}x + 4$

To find the area of triangle ABC, we can use the formula:

$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Using the coordinates:

• A (3, 5)
• B (-6, 2)
• C (-4, -4)

Plugging into the formula:

$\text{Area} = \frac{1}{2} \left| 3(2 - (-4)) + (-6)(-4 - 5) + (-4)(5 - 2) \right|$ $= \frac{1}{2} \left| 3(6) + (-6)(-9) + (-4)(3) \right|$ $= \frac{1}{2} \left| 18 + 54 - 12 \right|$ $= \frac{1}{2} \left| 60 \right|$ $= 30$

Thus, the area of triangle ABC is 30 square units.