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The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.?
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The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2)....
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The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2)....
Given Information
- Area of the triangle = 5
- Vertices = (2, 1) and (3, -2)
- The third vertex lies on the line y = x + 3
Finding Coordinates of the Third Vertex
Let the third vertex be (x, x + 3).
Area Formula for a Triangle
The area A of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula:
A = 1/2 * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Substituting Known Values
For our triangle, the vertices are:
- (x1, y1) = (2, 1)
- (x2, y2) = (3, -2)
- (x3, y3) = (x, x + 3)
Substituting into the area formula gives:
5 = 1/2 * | 2(-2 - (x + 3)) + 3((x + 3) - 1) + x(1 - (-2)) |
Simplifying the Equation
1. Simplify the expression inside the absolute value:
- 2(-2 - x - 3) = 2(-x - 5) = -2x - 10
- 3(x + 2) = 3x + 6
- x(3) = 3x
2. Combine the terms:
- -2x - 10 + 3x + 6 + 3x = 4x - 4
Thus, the area equation becomes:
10 = |4x - 4|
Solving for x
This gives two cases:
1. 4x - 4 = 10 → 4x = 14 → x = 3.5
2. 4x - 4 = -10 → 4x = -6 → x = -1.5
Finding y Coordinates
- For x = 3.5: y = 3.5 + 3 = 6.5 → Third vertex: (3.5, 6.5)
- For x = -1.5: y = -1.5 + 3 = 1.5 → Third vertex: (-1.5, 1.5)
Conclusion
The possible coordinates for the third vertex are (3.5, 6.5) or (-1.5, 1.5).
- Area of the triangle = 5
- Vertices = (2, 1) and (3, -2)
- The third vertex lies on the line y = x + 3
Finding Coordinates of the Third Vertex
Let the third vertex be (x, x + 3).
Area Formula for a Triangle
The area A of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula:
A = 1/2 * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Substituting Known Values
For our triangle, the vertices are:
- (x1, y1) = (2, 1)
- (x2, y2) = (3, -2)
- (x3, y3) = (x, x + 3)
Substituting into the area formula gives:
5 = 1/2 * | 2(-2 - (x + 3)) + 3((x + 3) - 1) + x(1 - (-2)) |
Simplifying the Equation
1. Simplify the expression inside the absolute value:
- 2(-2 - x - 3) = 2(-x - 5) = -2x - 10
- 3(x + 2) = 3x + 6
- x(3) = 3x
2. Combine the terms:
- -2x - 10 + 3x + 6 + 3x = 4x - 4
Thus, the area equation becomes:
10 = |4x - 4|
Solving for x
This gives two cases:
1. 4x - 4 = 10 → 4x = 14 → x = 3.5
2. 4x - 4 = -10 → 4x = -6 → x = -1.5
Finding y Coordinates
- For x = 3.5: y = 3.5 + 3 = 6.5 → Third vertex: (3.5, 6.5)
- For x = -1.5: y = -1.5 + 3 = 1.5 → Third vertex: (-1.5, 1.5)
Conclusion
The possible coordinates for the third vertex are (3.5, 6.5) or (-1.5, 1.5).
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The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.?
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The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.?.
The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area of a triangle is 5 . Two of its vertices are (2,1) & (3, -2). The third vertex is lies on y=x 3. Find the third vertex.?.
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