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Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).?
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Find the length of the altitudes from the vertices of the triangle who...
To find the length of the altitudes of a triangle, we need to find the equations of the lines that contain the altitudes and then use the equations to find the y-coordinates of the points where the altitudes intersect the sides of the triangle.
For the first altitude, we can use the coordinates (-1,1) and (5,2) to find the equation of the line. The slope of the line is (2-1)/(5-(-1)) = 3/6 = 1/2, so the equation of the line is y = (1/2)x + b. Substituting the coordinates (-1,1) into the equation, we find that b = 1. Therefore, the equation of the line containing the first altitude is y = (1/2)x + 1.
We can use the same process to find the equations of the lines containing the other two altitudes. The equation of the line containing the second altitude is y = (2/3)x + (4/3), and the equation of the line containing the third altitude is y = (-1/2)x + (5/2).
To find the y-coordinates of the points where the altitudes intersect the sides of the triangle, we can substitute the x-coordinates of the vertices into the equations of the lines. For the first altitude, the x-coordinate is -1, so the y-coordinate is (-1/2)*(-1) + 1 = 1/2. For the second altitude, the x-coordinate is 5, so the y-coordinate is (2/3)*5 + (4/3) = 8/3. For the third altitude, the x-coordinate is 3, so the y-coordinate is (-1/2)*3 + (5/2) = 1/2.
The lengths of the altitudes are then the distances between the y-coordinates of the vertices and the y-coordinates of the points where the altitudes intersect the sides of the triangle. The first altitude has length |1 - (1/2)| = 1/2, the second altitude has length |2 - (8/3)| = (4/3), and the third altitude has length |(-1) - (1/2)| = 3/2. Therefore, the lengths of the altitudes are 1/2, 4/3, and 3/2.
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Find the length of the altitudes from the vertices of the triangle who...
To find the lengths of the altitudes from the vertices of the triangle with vertices A(-1,1), B(5,2), and C(3,-1), follow these detailed steps:
Step 1: Calculate the Area of the Triangle
- Use the formula for the area of a triangle given vertices (x1, y1), (x2, y2), (x3, y3):
Area = 1/2 * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
- Substitute the coordinates:
Area = 1/2 * | -1(2 - (-1)) + 5(-1 - 1) + 3(1 - 2) |
Area = 1/2 * | -1(3) + 5(-2) + 3(-1) |
Area = 1/2 * | -3 - 10 - 3 |
Area = 1/2 * | -16 | = 8 square units
Step 2: Calculate the Lengths of the Sides
- Use the distance formula between two points (x1, y1) and (x2, y2):
Distance = √((x2 - x1)² + (y2 - y1)²)
- Calculate side lengths:
- AB = √((5 - (-1))² + (2 - 1)²) = √(36 + 1) = √37
- BC = √((3 - 5)² + (-1 - 2)²) = √(4 + 9) = √13
- CA = √((-1 - 3)² + (1 - (-1))²) = √(16 + 4) = √20
Step 3: Find the Altitudes
- Use the formula for altitude (h) from vertex to opposite side:
h = (2 * Area) / base
- Calculate each altitude:
- Altitude from A (h_a) = (2 * 8) / BC = 16 / √13
- Altitude from B (h_b) = (2 * 8) / CA = 16 / √20
- Altitude from C (h_c) = (2 * 8) / AB = 16 / √37
Final Results
- Altitude from A: 16 / √13
- Altitude from B: 16 / √20
- Altitude from C: 16 / √37
Thus, the lengths of the altitudes from vertices A, B, and C are 16 / √13, 16 / √20, and 16 / √37 respectively.
Step 1: Calculate the Area of the Triangle
- Use the formula for the area of a triangle given vertices (x1, y1), (x2, y2), (x3, y3):
Area = 1/2 * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
- Substitute the coordinates:
Area = 1/2 * | -1(2 - (-1)) + 5(-1 - 1) + 3(1 - 2) |
Area = 1/2 * | -1(3) + 5(-2) + 3(-1) |
Area = 1/2 * | -3 - 10 - 3 |
Area = 1/2 * | -16 | = 8 square units
Step 2: Calculate the Lengths of the Sides
- Use the distance formula between two points (x1, y1) and (x2, y2):
Distance = √((x2 - x1)² + (y2 - y1)²)
- Calculate side lengths:
- AB = √((5 - (-1))² + (2 - 1)²) = √(36 + 1) = √37
- BC = √((3 - 5)² + (-1 - 2)²) = √(4 + 9) = √13
- CA = √((-1 - 3)² + (1 - (-1))²) = √(16 + 4) = √20
Step 3: Find the Altitudes
- Use the formula for altitude (h) from vertex to opposite side:
h = (2 * Area) / base
- Calculate each altitude:
- Altitude from A (h_a) = (2 * 8) / BC = 16 / √13
- Altitude from B (h_b) = (2 * 8) / CA = 16 / √20
- Altitude from C (h_c) = (2 * 8) / AB = 16 / √37
Final Results
- Altitude from A: 16 / √13
- Altitude from B: 16 / √20
- Altitude from C: 16 / √37
Thus, the lengths of the altitudes from vertices A, B, and C are 16 / √13, 16 / √20, and 16 / √37 respectively.
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Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).?
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Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).?.
Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the length of the altitudes from the vertices of the triangle whose vertices are (-1,1),(5,2)and(3,-1).?.
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