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 = (y_{2} - y_{1})/(x_{2} - x_{1})
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 90°. 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 :
Part 1 :
Slope of AB = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (x_{1}, y_{1}) = (1, -4) and (x_{2}, y_{2}) = (2, -3).
= (-3 + 4)/(2 - 1)
= 1/1
= 1
Slope of BC = (-7 + 3)/(4 - 2)
= -4/2
= -2
Slope of AC = (-7 + 4)/(4 - 1)
= -3/3
= -1
Slope of AB x Slope of AC = (-1)(1)
= -1
AB is perpendicular to AC, ∠A = 90°.
The vertices A, B and C form a right triangle.
Part 2 :
Using the distance formula d = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{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 = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]
Substitute (x_{1}, y_{1}) = (1, -4) and (x_{2}, y_{2}) = (2, -3).
AB = √[(2 - 1)^{2} + (-3 + 4)^{2}]
= √[1^{2} + 1^{2}]
= √[1 + 1]
= √2
AB^{2} = 2
BC = √[(4 - 2)^{2} + (-7 + 3)^{2}] = √20
BC^{2} = 20
AC = √[(4 - 1)^{2} + (-7 + 4)^{2}] = √18
AC^{2} = 18
20 = 2 + 18 ----> BC^{2} = AB^{2} + AC^{2}
The vertices A, B and C satisfy Pythagorean theorem.
Example 2 :
L(0, 5), M(9, 12), N(3, 14)
Solution :
Part 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 x Slope of LN = (-1/3)(3)
= -1
MN is perpendicular to LN, ∠N = 90°.
The vertices L, M and N form a right triangle.
Part 2 :
The vertices are L(0, 5), M(9, 12) and N(3, 14).
LM = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]
Substitute (x_{1}, y_{1}) = (0, 5) and (x_{2}, y_{2}) = (9, 12).
LM = √[(9 - 0)^{2} + (12 - 5)^{2}]
= √[9^{2} + 7^{2}]
= √[81 + 49]
= √130
LM^{2} = 130
MN = √[(3 - 9)^{2} + (14 - 12)^{2}] = √40
MN^{2} = 40
LN = √[(3 - 0)^{2} + (14 - 5)^{2}] = √90
LN^{2} = 90
130 = 40 + 90 ----> LM^{2} = MN^{2} + LN^{2}
The vertices L, M and N satisfy Pythagorean theorem.
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