Question

In ΔABC, ∠ABC = 90°, A(6, − 7), B(−3, 5) and BC = 20 units. Find the length of AB and AC.

Answer 1

Given:

ΔABC, ∠ABC = 90°

Coordinates: A(6, −7), B(−3, 5)

Length BC = 20 units. Find the lengths of AB and AC.

Step 1: Calculate length AB

`"Distance" = sqrt((x_2 − x_1)^2 + (y_2 − y_1)^2)`    ...[Using the distance formula for points]

Substitute A(6, −7), and B(−3, 5)

\[AB = \sqrt{(-3 - 6)^2 + (5 - (-7))^2}\]

= \[\sqrt{(-9)^2 + (12)^2}\] 

= \[\sqrt{81 + 144}\] 

= \[\sqrt{225}\]

= 15

Step 2: Use the Pythagorean theorem

Since ∠ABC = 90°, AB and BC are the legs of the right triangle, and AC is the hypotenuse.

AC² = AB² + BC² ... [According to the Pythagorean theorem]

AB = 15, BC = 20

\[AC = \sqrt{(15)^2 + (20)^2}\]

= \[\sqrt{225 + 400}\]

= \[\sqrt{625}\]

= 25

AC = 25 units

AB = 15 units and AC = 25 units