Anonymous
Sep 3
Prove that a triangle with vertices A(3, -6), B(8, -3), and C(-1, -1) is a right-angled triangle. Find its area as well.
Show whether is a right angled triangle and it's area
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ChatGPT
Sep 3
To prove that the triangle ABC is a right-angled triangle, we can calculate the slopes of the sides AB, BC, and AC.
The slope of AB = (y2 - y1) / (x2 - x1) = (-3 - (-6)) / (8 - 3) = 3 / 5
The slope of BC = (-1 - (-3)) / (-1 - 8) = 2 / 9
The slope of AC = (-1 - (-6)) / (-1 - 3) = 5 / 4
Since the product of the slopes of two perpendicular lines is -1, we can see that the slopes of AB and AC are negative reciprocals of each other. Therefore, triangle ABC is a right-angled triangle with right angle at vertex B.
To find the area of the triangle, we can use the formula for the area of a triangle given its vertices:
Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
= 1/2 * |3(-1 - (-1)) + 8(-1 - (-6)) + (-1)(-6 - (-3))|
= 1/2 * |3(0) + 8(5) + (-1)(-3)|
= 1/2 * |0 + 40 + 3|
= 1/2 * 43
= 21.5
Therefore, the area of triangle ABC is 21.5 square units.
The slope of AB = (y2 - y1) / (x2 - x1) = (-3 - (-6)) / (8 - 3) = 3 / 5
The slope of BC = (-1 - (-3)) / (-1 - 8) = 2 / 9
The slope of AC = (-1 - (-6)) / (-1 - 3) = 5 / 4
Since the product of the slopes of two perpendicular lines is -1, we can see that the slopes of AB and AC are negative reciprocals of each other. Therefore, triangle ABC is a right-angled triangle with right angle at vertex B.
To find the area of the triangle, we can use the formula for the area of a triangle given its vertices:
Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
= 1/2 * |3(-1 - (-1)) + 8(-1 - (-6)) + (-1)(-6 - (-3))|
= 1/2 * |3(0) + 8(5) + (-1)(-3)|
= 1/2 * |0 + 40 + 3|
= 1/2 * 43
= 21.5
Therefore, the area of triangle ABC is 21.5 square units.
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Google Gemini AI
Sep 3
Proof of Right Angle:
Using the distance formula, we can calculate the lengths of the three sides of the triangle:
- AB = √[(8 - 3)^2 + (-3 - (-6))^2] = √(25 + 9) = 5
- BC = √[(-1 - 8)^2 + (-1 - (-3))^2] = √(81 + 4) = √85
- AC = √[(-1 - 3)^2 + (-1 - (-6))^2] = √(16 + 25) = √41
By the Pythagorean theorem, a triangle is right-angled if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides. In this case, BC is the longest side.
BC^2 = 85
AB^2 + AC^2 = 25 + 41 = 66
Since BC^2 = AB^2 + AC^2, the triangle is right-angled.
Area of the Triangle:
The area of a right-angled triangle can be calculated using the formula:
Area = (1/2) - base - height
Let's take AB as the base and AC as the height.
Area = (1/2) - 5 - √41
Area = 10.25√41 square units
Using the distance formula, we can calculate the lengths of the three sides of the triangle:
- AB = √[(8 - 3)^2 + (-3 - (-6))^2] = √(25 + 9) = 5
- BC = √[(-1 - 8)^2 + (-1 - (-3))^2] = √(81 + 4) = √85
- AC = √[(-1 - 3)^2 + (-1 - (-6))^2] = √(16 + 25) = √41
By the Pythagorean theorem, a triangle is right-angled if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides. In this case, BC is the longest side.
BC^2 = 85
AB^2 + AC^2 = 25 + 41 = 66
Since BC^2 = AB^2 + AC^2, the triangle is right-angled.
Area of the Triangle:
The area of a right-angled triangle can be calculated using the formula:
Area = (1/2) - base - height
Let's take AB as the base and AC as the height.
Area = (1/2) - 5 - √41
Area = 10.25√41 square units
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