Question

The trangie ABC has vertices A(1,3),B(3,7) and C(−1,9).

(i) Find the gradient of AB. (ii) Hence show that angle ABC is a right angle.

Answer 1

The gradient of AB is 2, and the gradient of BC is -1/2. The product of these gradients is -1, so angle ABC is a right angle.

Step-by-step explanation:

The gradient of a line is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points. For points A(1,3) and B(3,7), the gradient (m) is (7 - 3) / (3 - 1) = 4 / 2 = 2. To show that angle ABC is a right angle, we need to determine the gradient of BC and show that the product of the gradients of AB and BC is -1, as perpendicular lines have gradients that are negative reciprocals of each other.

For points B(3,7) and C(-1,9), the gradient of BC is (9 - 7) / (-1 - 3) = 2 / -4 = -1/2. Multiplying the gradient of AB with the gradient of BC gives 2 * (-1/2) = -1, which confirms that lines AB and BC are perpendicular, hence, angle ABC is a right angle.