Question

Show that the given points form a right angled triangle and check whether they satisfy Pythagoras theorem.

L(0, 5), M(9, 12) and N(3, 14)

Answer 1

The vertices are L(0, 5), M(9, 12) and N(3, 14)

Slope of a line = `(y_2 - y_1)/(x_2 - x_1)`

Slope of LM = `(12 - 5)/(9 - 0) = 7/9`

Slope of MN = `(14 - 12)/(3 - 9) = 2/(-6) = -1/3`

Slope of LN = `(14 - 5)/(3 - 0) = 9/3` = 3

Slope of MN × Slope of LN = `-1/3 xx 3` = –1

∴ MN ⊥ LN

∠N = 90°

∴ LMN is a right angle triangle.

Verification:

Distance = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

LN = `sqrt((3 - 0)^2 + (14 - 5)^2`

= `sqrt(3^2 + 9^2)`

= `sqrt(9 + 81)`

= `sqrt(90)`

MN = `sqrt((9 - 3)^2 + (12 - 14)^2`

= `sqrt(6^2+ (- 2)^2`

= `sqrt(36 + 4)`

= `sqrt(40)`

LM = `sqrt((9 - 0)^2 + (12 - 5)^2`

= `sqrt(9^2 + 7^2)`

= `sqrt(81 + 49)`

= `sqrt(130)`

LM² = LN² + MN²

130 = 90 + 40

130 = 130

⇒ Pythagoras theorem is verified.