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The points (1, 2 , 3) , (- 1, -1 , - 1) and (3, 5 , 7) are the vertices of
- a)an isosceles triangle
- b)a right triangle
- c)an equilateral triangle
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The points (1, 2 , 3) , (- 1, -1 , - 1) and (3, 5 , 7) are the vertice...
Explanation:
To determine the nature of the triangle formed by the given points, we need to calculate the lengths of all three sides of the triangle.
Step 1: Calculate the lengths of the sides
Let's calculate the lengths of the sides using the distance formula:
Side AB:
√[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
= √[(-1 - 1)² + (-1 - 2)² + (-1 - 3)²]
= √[(-2)² + (-3)² + (-4)²]
= √[4 + 9 + 16]
= √29
Side BC:
√[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
= √[(3 - (-1))² + (5 - (-1))² + (7 - (-1))²]
= √[(3 + 1)² + (5 + 1)² + (7 + 1)²]
= √[4² + 6² + 8²]
= √[16 + 36 + 64]
= √116
= 2√29
Side AC:
√[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
= √[(3 - 1)² + (5 - 2)² + (7 - 3)²]
= √[2² + 3² + 4²]
= √[4 + 9 + 16]
= √29
Step 2: Analyze the lengths of the sides
From the calculations above, we can see that all three sides of the triangle have the same length, which is √29.
Step 3: Determine the nature of the triangle
Based on the lengths of the sides, we can conclude that the triangle is an isosceles triangle because it has two sides of equal length (AB and AC).
Therefore, the correct answer is option D: none of these because the given triangle is not a right triangle, equilateral triangle, or a scalene triangle.
To determine the nature of the triangle formed by the given points, we need to calculate the lengths of all three sides of the triangle.
Step 1: Calculate the lengths of the sides
Let's calculate the lengths of the sides using the distance formula:
Side AB:
√[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
= √[(-1 - 1)² + (-1 - 2)² + (-1 - 3)²]
= √[(-2)² + (-3)² + (-4)²]
= √[4 + 9 + 16]
= √29
Side BC:
√[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
= √[(3 - (-1))² + (5 - (-1))² + (7 - (-1))²]
= √[(3 + 1)² + (5 + 1)² + (7 + 1)²]
= √[4² + 6² + 8²]
= √[16 + 36 + 64]
= √116
= 2√29
Side AC:
√[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
= √[(3 - 1)² + (5 - 2)² + (7 - 3)²]
= √[2² + 3² + 4²]
= √[4 + 9 + 16]
= √29
Step 2: Analyze the lengths of the sides
From the calculations above, we can see that all three sides of the triangle have the same length, which is √29.
Step 3: Determine the nature of the triangle
Based on the lengths of the sides, we can conclude that the triangle is an isosceles triangle because it has two sides of equal length (AB and AC).
Therefore, the correct answer is option D: none of these because the given triangle is not a right triangle, equilateral triangle, or a scalene triangle.
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The points (1, 2 , 3) , (- 1, -1 , - 1) and (3, 5 , 7) are the vertices ofa)an isosceles triangleb)a right trianglec)an equilateral triangled)none of theseCorrect answer is option 'D'. Can you explain this answer?
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