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A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal. Show that p/q = cosβ - cos𝛼/ sin𝛼 - sinβ

Solution:

Given, a ladder rests against a vertical wall at an inclination α to the horizontal.

Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal.

We have to prove that p/q = cosβ - cos𝛼/ sin𝛼 - sinβ.

Let AB be the ladder resting against a vertical wall

Angle of inclination, ∠BAO = 𝛼

When the foot of the ladder is pulled away through a distance p, the upper end of the ladder slides a distance q down the wall.

After sliding, SQ is the ladder

The pulled away distance, SA = p

Distance slide down the wall, BQ = q

Angle of inclination, ∠QSO = β

Let us consider,

QO = x

AO = y

Now, height of the wall,BO = q + x

Considering triangle BAO, we get

cos𝛼 = OA/AB

cos𝛼 = y/AB

y = ABcos𝛼

Also, OA = AB cos𝛼 --------------- (1)

sin𝛼 = OB/AB

OB = BA sin𝛼 ----------------------- (2)

Considering triangle QSO, we get,

sinβ = QO/SQ

QO = SQ sinβ

cosβ = SO/SQ

SO = SQcosβ

Since the length of the ladder remains the same, we have,

AB = SQ

Hence,

QO = AB sinβ ---------------------- (3)

SO = AB cosβ ---------------------- (4)

We know that,

SA = OS - AO

p = AB cosβ - AB cos𝛼

p = AB ( cosβ - cos𝛼) ------------------- (5)

BQ = BO - QO

q = BA sin𝛼 - AB sinβ

q = BA (sin𝛼 - sinβ) ----------------------- (6)

According to the question, we need to prove that p/q = cosβ - cos𝛼/ sin𝛼 - sinβ.

p/q = AB ( cosβ - cos𝛼)/BA (sin𝛼 - sinβ)

p/q = (cosβ - cos𝛼)/ (sin𝛼 - sinβ)

Therefore,p/q = (cosβ - cos𝛼)/ (sin𝛼 - sinβ)

✦ Try This: A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance ‘a’, so that it slides a distance ‘b’ down the wall, making an angle β with the horizontal. The value of a/b is

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 15

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal. Show that p/q = cosβ - cos𝛼/ sin𝛼 - sinβ

Summary:

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal. It is shown that p/q = cosβ - cos𝛼/ sin𝛼 - sinβ

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