Question

In ΔABC, ∠ABC = 90° C = (2, 0) and B = (−2, 3). If AC = 13 units, find the lengths of BC and AB.

Answer 1

Given

∠ABC = 90° coordinates of C and B, and AC = 13 units.

finding the lengths of BC and AB.

Step 1: Use distance formula to find BC

BC = `sqrt((x_2 − x_1)^2 + (y_2 − y_1)^2)`

Substitute B = (−2, 3) and C = (2, 0)

BC = `sqrt((−2 − 2)^2 + (3 − 0)^2)`

BC = `sqrt((−4)^2 + (3)^2)`

BC = `sqrt(16 + 9)`

BC = `sqrt25`

BC = 5 units

Step 2: Use Pythagoras theorem to find AB

Since ∠ABC = 90°, triangle ABC is a right-angled triangle at B.

By Pythagoras Theorem:

AC² = AB² + BC²

Given AC = 13, BC = 5

13² = AB² + 5²

169 = AB² + 25

AB² = 144

AB = `sqrt144​`

AB = 12 units