Class 10 Exam > Class 10 Questions > A ladder rests against a wall at an angle alp...
Start Learning for Free
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.?
Most Upvoted Answer
A ladder rests against a wall at an angle alpha to the horizontal. Its...
Community Answer
A ladder rests against a wall at an angle alpha to the horizontal. Its...
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
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
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.?
Question Description
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.?.
Solutions 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.? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of 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.? defined & explained in the simplest way possible. Besides giving the explanation of
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 detailed solution 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.? has been provided alongside types of 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.? theory, EduRev gives you an
ample number of questions to practice 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.? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup