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A triangle with vertices (4, 0), (- 1, - 1) and (3, 5) is a/an
- a)equilateral triangle
- b)right-angled triangle
- c)isosceles right-angled triangle
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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A triangle with vertices (4, 0), (- 1, - 1) and (3, 5) is a/ana)equila...
A(4, 0), B(-1, -1), C(3, 5) are the vertices of a triangle. Then
Thus, the triangle is isosceles and right angle triangle.
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A triangle with vertices (4, 0), (- 1, - 1) and (3, 5) is a/ana)equila...
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A triangle with vertices (4, 0), (- 1, - 1) and (3, 5) is a/ana)equila...
To determine whether the given triangle is an isosceles right-angled triangle or not, we can use the distance formula and the Pythagorean theorem.
Step 1: Find the lengths of the three sides of the triangle
Let's label the vertices of the triangle as A(4, 0), B(-1, -1), and C(3, 5).
The distance formula is given by:
Distance between two points (x1, y1) and (x2, y2) = √[(x2-x1)^2 + (y2-y1)^2]
Using this formula, we can find the lengths of the three sides of the triangle:
Side AB: √[(-1-4)^2 + (-1-0)^2] = √[25 + 1] = √26
Side BC: √[(3-(-1))^2 + (5-(-1))^2] = √[16 + 36] = √52 = 2√13
Side AC: √[(3-4)^2 + (5-0)^2] = √[1 + 25] = √26
Step 2: Check if the triangle is isosceles
An isosceles triangle has two sides of equal length. In this case, AB and AC both have a length of √26. So, the triangle is isosceles.
Step 3: Check if the triangle is right-angled
A right-angled triangle has one angle measuring 90 degrees. To check if the triangle is right-angled, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the longest side is AC, so we can check if AC^2 = AB^2 + BC^2:
AC^2 = (√26)^2 = 26
AB^2 + BC^2 = (√26)^2 + (2√13)^2 = 26 + 4*13 = 26 + 52 = 78
Since AC^2 is not equal to AB^2 + BC^2, the triangle is not right-angled.
Conclusion
Therefore, the given triangle with vertices (4, 0), (-1, -1), and (3, 5) is an isosceles triangle, but it is not right-angled. Hence, the correct answer is option 'C' - isosceles right-angled triangle.
Step 1: Find the lengths of the three sides of the triangle
Let's label the vertices of the triangle as A(4, 0), B(-1, -1), and C(3, 5).
The distance formula is given by:
Distance between two points (x1, y1) and (x2, y2) = √[(x2-x1)^2 + (y2-y1)^2]
Using this formula, we can find the lengths of the three sides of the triangle:
Side AB: √[(-1-4)^2 + (-1-0)^2] = √[25 + 1] = √26
Side BC: √[(3-(-1))^2 + (5-(-1))^2] = √[16 + 36] = √52 = 2√13
Side AC: √[(3-4)^2 + (5-0)^2] = √[1 + 25] = √26
Step 2: Check if the triangle is isosceles
An isosceles triangle has two sides of equal length. In this case, AB and AC both have a length of √26. So, the triangle is isosceles.
Step 3: Check if the triangle is right-angled
A right-angled triangle has one angle measuring 90 degrees. To check if the triangle is right-angled, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the longest side is AC, so we can check if AC^2 = AB^2 + BC^2:
AC^2 = (√26)^2 = 26
AB^2 + BC^2 = (√26)^2 + (2√13)^2 = 26 + 4*13 = 26 + 52 = 78
Since AC^2 is not equal to AB^2 + BC^2, the triangle is not right-angled.
Conclusion
Therefore, the given triangle with vertices (4, 0), (-1, -1), and (3, 5) is an isosceles triangle, but it is not right-angled. Hence, the correct answer is option 'C' - isosceles right-angled triangle.
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A triangle with vertices (4, 0), (- 1, - 1) and (3, 5) is a/ana)equilateral triangleb)right-angled trianglec)isosceles right-angled triangled)none of theseCorrect answer is option 'C'. Can you explain this answer?
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