SAT Exam > SAT Questions > The three points A (7, 0, 10), B (6, -1, 6), ...
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The three points A (7, 0, 10), B (6, -1, 6), C (9, -4, 6) form ________.
- a)equilateral triangle
- b)right angled triangle
- c)isosceles triangle
- d)right angled isosceles triangle
Correct answer is option 'D'. Can you explain this answer?
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The three points A (7, 0, 10), B (6, -1, 6), C (9, -4, 6) form _______...
Explanation:
Right Angled Isosceles Triangle:
- A right-angled isosceles triangle is a triangle with one right angle (90 degrees) and two equal sides.
- In this case, we can calculate the distances between the points to determine the type of triangle formed.
Calculating Distances:
- The distance formula between two points (x1, y1, z1) and (x2, y2, z2) in 3D space is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance AB:
- AB = √((6 - 7)^2 + (-1 - 0)^2 + (6 - 10)^2)
- AB = √(1 + 1 + 16)
- AB = √18
Distance BC:
- BC = √((9 - 6)^2 + (-4 - (-1))^2 + (6 - 6)^2)
- BC = √(9 + 9 + 0)
- BC = √18
Distance AC:
- AC = √((9 - 7)^2 + (-4 - 0)^2 + (6 - 10)^2)
- AC = √(4 + 16 + 16)
- AC = √36
- AC = 6
Conclusion:
- Since AB = BC = √18 and AC = 6, the triangle formed by the points A, B, and C is a right-angled isosceles triangle.
- Therefore, the correct answer is option 'D'.
Right Angled Isosceles Triangle:
- A right-angled isosceles triangle is a triangle with one right angle (90 degrees) and two equal sides.
- In this case, we can calculate the distances between the points to determine the type of triangle formed.
Calculating Distances:
- The distance formula between two points (x1, y1, z1) and (x2, y2, z2) in 3D space is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance AB:
- AB = √((6 - 7)^2 + (-1 - 0)^2 + (6 - 10)^2)
- AB = √(1 + 1 + 16)
- AB = √18
Distance BC:
- BC = √((9 - 6)^2 + (-4 - (-1))^2 + (6 - 6)^2)
- BC = √(9 + 9 + 0)
- BC = √18
Distance AC:
- AC = √((9 - 7)^2 + (-4 - 0)^2 + (6 - 10)^2)
- AC = √(4 + 16 + 16)
- AC = √36
- AC = 6
Conclusion:
- Since AB = BC = √18 and AC = 6, the triangle formed by the points A, B, and C is a right-angled isosceles triangle.
- Therefore, the correct answer is option 'D'.
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The three points A (7, 0, 10), B (6, -1, 6), C (9, -4, 6) form _______...
We know, distance between two points (x1, y1, z1) and (x2, y2, z2) is 
Since AB = BC and AB2+BC2 = AC2 so, it forms right angled isosceles triangle.
Since AB = BC and AB2+BC2 = AC2 so, it forms right angled isosceles triangle.
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