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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)
Solution
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.
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