Without using the distance formula, show that the points A (4, –2), B (–4, 4) and C (10, 6) are the vertices of a right-angled triangle.
Asked by tusharparik • 14 days ago
Without using the distance formula, show that the points A (4, –2), B (–4, 4) and C (10, 6) are the vertices of a right-angled triangle.
Answers
Mathematics - Right-Angled Triangle Without Distance Formula
To prove that points A(4, -2), B(-4, 4), and C(10, 6) are vertices of a right-angled triangle without using the distance formula, we can use the concept of slopes.
Concept of Slopes
In coordinate geometry, the slope of a line joining two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
Two lines are perpendicular if the product of their slopes is -1.
Steps to Prove Right-Angled Triangle
- Calculate the slopes of lines AB, BC, and AC.
- Check if the product of any two slopes is -1. If so, the corresponding lines are perpendicular, and the triangle is right-angled.
Calculations
- Slope of AB (mAB):
mAB = (4 - (-2)) / (-4 - 4) = (4 + 2) / (-8) = 6 / -8 = -3/4 - Slope of BC (mBC):
mBC = (6 - 4) / (10 - (-4)) = 2 / (10 + 4) = 2 / 14 = 1/7 - Slope of AC (mAC):
mAC = (6 - (-2)) / (10 - 4) = (6 + 2) / 6 = 8 / 6 = 4/3
Checking for Perpendicularity
Now, we need to check if the product of any two slopes is -1:
mAB * mBC = (-3/4) * (1/7) = -3/28 (Not -1)
mAB * mAC = (-3/4) * (4/3) = -1 (Perpendicular)
mBC * mAC = (1/7) * (4/3) = 4/21 (Not -1)
Conclusion
Since the product of the slopes of AB and AC is -1, AB and AC are perpendicular. Therefore, triangle ABC is a right-angled triangle with the right angle at vertex A.
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