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A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.?
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Introduction:
We need to prove that a/b = (cos β - cos α) / (sin α - sin β), where a is the distance the foot of the ladder is pulled away from the wall, b is the distance it slides down the wall, α is the angle the ladder makes with the horizontal initially, and β is the angle the ladder makes with the horizontal after sliding down.
Proof:
To prove the given equation, we will consider a right-angled triangle formed by the ladder, the wall, and the ground. Let's label the sides of the triangle as follows:
- The hypotenuse of the triangle (the ladder) is labeled as L.
- The side opposite to angle α is labeled as a.
- The side adjacent to angle α is labeled as h.
- The side opposite to angle β is labeled as b.
- The side adjacent to angle β is labeled as d.
Using Trigonometric Ratios:
1. Using trigonometric ratios in the right-angled triangle, we have:
- sin α = a / L (opposite / hypotenuse)
- cos α = h / L (adjacent / hypotenuse)
- sin β = b / L (opposite / hypotenuse)
- cos β = d / L (adjacent / hypotenuse)
2. Rearranging the equations, we get:
- a = L * sin α
- h = L * cos α
- b = L * sin β
- d = L * cos β
Applying Pythagoras Theorem:
3. Using Pythagoras theorem in the right-angled triangle, we have:
- h² + a² = L² (by substituting the values of h and a from step 2)
4. After sliding down the wall, the right-angled triangle changes, and we have:
- d² + b² = L² (by substituting the values of d and b from step 2)
Simplifying the Equations:
5. Subtracting equation 4 from equation 3, we get:
- h² - d² + a² - b² = 0
6. Rearranging the terms, we have:
- (h² - d²) + (a² - b²) = 0
7. Factoring the difference of squares, we get:
- (h + d)(h - d) + (a + b)(a - b) = 0
8. Dividing both sides by (h + d), we get:
- (h - d) / (h + d) = -(a + b)(a - b) / (h + d)
Substituting Trigonometric Ratios:
9. Substituting the values of h, d, a, and b from step 2, we get:
- (L * cos α - L * cos β) / (L * cos α + L * cos β) = -[(L * sin α + L * sin β)(L * sin α - L * sin β)] / (L * cos α + L * cos β)
10. Simplifying the equation, we get:
- (cos
We need to prove that a/b = (cos β - cos α) / (sin α - sin β), where a is the distance the foot of the ladder is pulled away from the wall, b is the distance it slides down the wall, α is the angle the ladder makes with the horizontal initially, and β is the angle the ladder makes with the horizontal after sliding down.
Proof:
To prove the given equation, we will consider a right-angled triangle formed by the ladder, the wall, and the ground. Let's label the sides of the triangle as follows:
- The hypotenuse of the triangle (the ladder) is labeled as L.
- The side opposite to angle α is labeled as a.
- The side adjacent to angle α is labeled as h.
- The side opposite to angle β is labeled as b.
- The side adjacent to angle β is labeled as d.
Using Trigonometric Ratios:
1. Using trigonometric ratios in the right-angled triangle, we have:
- sin α = a / L (opposite / hypotenuse)
- cos α = h / L (adjacent / hypotenuse)
- sin β = b / L (opposite / hypotenuse)
- cos β = d / L (adjacent / hypotenuse)
2. Rearranging the equations, we get:
- a = L * sin α
- h = L * cos α
- b = L * sin β
- d = L * cos β
Applying Pythagoras Theorem:
3. Using Pythagoras theorem in the right-angled triangle, we have:
- h² + a² = L² (by substituting the values of h and a from step 2)
4. After sliding down the wall, the right-angled triangle changes, and we have:
- d² + b² = L² (by substituting the values of d and b from step 2)
Simplifying the Equations:
5. Subtracting equation 4 from equation 3, we get:
- h² - d² + a² - b² = 0
6. Rearranging the terms, we have:
- (h² - d²) + (a² - b²) = 0
7. Factoring the difference of squares, we get:
- (h + d)(h - d) + (a + b)(a - b) = 0
8. Dividing both sides by (h + d), we get:
- (h - d) / (h + d) = -(a + b)(a - b) / (h + d)
Substituting Trigonometric Ratios:
9. Substituting the values of h, d, a, and b from step 2, we get:
- (L * cos α - L * cos β) / (L * cos α + L * cos β) = -[(L * sin α + L * sin β)(L * sin α - L * sin β)] / (L * cos α + L * cos β)
10. Simplifying the equation, we get:
- (cos
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A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.?
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A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.?.
A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ladder rests against a wall at an angle alpha to the horizontal. Its foot is pulled away from the wall through a distance a , so that it slides distance b down the wall, making an angle beta with horizontal. Then show that a/b = cos beta - cos alpha / sin alpha - sin beta.?.
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