HOW TO PROVE THE GIVEN VERTICES FORM A RIGHT TRIANGLE USING SLOPE
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The following steps would be useful to prove that given vertices form a right triangle.
Step 1 :
Use slope formula to find the slope of each side of the triangle.
Slope = β½ΚΈ2 β» ΚΈ1βΎβββ2 β β1β
Step 2 :
Check whether the product of slopes of any two sides is equal to -1.
If the product of slopes of any two sides is equal to -1, then those two sides are perpendicular and the angle between them is 90Β°. Then the given vertices form a right triangle, other wise they don't.
In each case, show that the given vertices form a right triangle and check whether they satisfy Pythagorean Theorem.
Example 1 :
A(1, -4), B(2, -3), C(4,-7)
Solution :
Slope of AB = β½β»Β³ βΊ β΄βΎβββ β ββ
= ΒΉββ
= 1
Slope of BC = β½β»β· βΊ Β³βΎβββ β ββ
= -β΄ββ
= -2
Slope of AC = β½β»β· βΊ β΄βΎβββ β ββ
= -Β³ββ
= -1
Slope of AB x Slope of AC = (-1)(1)
= -1
AB is perpendicular to AC, β A = 90Β°.
Using the distance formula d = β[(x2 - x1)2 + (y2 - y1)2], find the length of each side of the triangle.
Then, Check Pythagorean Theorem for the lengths.
That is, square of the larger side has to be equal to sum of the squares of other two sides.
The vertices are A(1, -4), B(2, -3) and C(4,-7).
AB = β[(x2 - x1)2 + (y2 - y1)2]
Substitute (x1, y1) = (1, -4) and (x2, y2) = (2, -3).
AB = β[(2 - 1)2 + (-3 + 4)2]
= β[12 + 12]
= β[1 + 1]
= β2
AB2 = 2
BC = β[(4 - 2)2 + (-7 + 3)2] = β20
BC2 = 20
AC = β[(4 - 1)2 + (-7 + 4)2] = β18
AC2 = 18
20 = 2 + 18 ----> BC2 = AB2 + AC2
The points A, B and C satisfy Pythagorean theorem.
Therefore, the points A, B and C form a right triangle.
Example 2 :
L(0, 5), M(9, 12), N(3, 14)
Solution :
Slope of LM = β½ΒΉΒ² β» β΅βΎβββ β ββ
= β·ββ
Slope of MN = β½ΒΉβ΄ β» ΒΉΒ²βΎβββ β ββ
= Β²βββ
= -β
Slope of LN = β½ΒΉβ΄ β» β΅βΎβββ β ββ
= βΉββ
= 3
Slope of MN x Slope of LN = (-β
)(3)
= -1
MN is perpendicular to LN, β N = 90Β°.
Using the distance formula d = β[(x2 - x1)2 + (y2 - y1)2], find the length of each side of the triangle.
Then, Check Pythagorean Theorem for the lengths.
That is, square of the larger side has to be equal to sum of the squares of other two sides.
LM = β[(x2 - x1)2 + (y2 - y1)2]
Substitute (x1, y1) = (0, 5) and (x2, y2) = (9, 12).
LM = β[(9 - 0)2 + (12 - 5)2]
= β[92 + 72]
= β[81 + 49]
= β130
LM2 = 130
MN = β[(3 - 9)2 + (14 - 12)2] = β40
MN2 = 40
LN = β[(3 - 0)2 + (14 - 5)2] = β90
LN2 = 90
130 = 40 + 90 ----> LM2 = MN2 + LN2
The points L, M and N satisfy Pythagorean theorem.
Therefore, the points L, M and N form a right triangle.
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