Question

An 18 meters ladder is resting against a wall and makes an angle of 30° with the ground.

If the ladder is pushed vertically along the wall by a distance of x meters, the angle is
changed to 60° with the ground. The value of x is
A)9(√3 −1)
B) 2√3
c) ³(√3-1)
D) 9-√3

Answer 1

A

Explanation:

using the sine ratio and the exact values

sin30° =
`(1)/(2)` and sin60° =
`(√(3) )/(2)`

calculate the height of the wall below x , call it h

the hypotenuse is the length of the ladder

using the sine ratio in the lower right triangle.

sin30° =
`(opposite)/(hypotenuse)` =
`(h)/(18)` =
`(1)/(2)` ( cross- multiply )

2h = 18 ( divide both sides by 2 )

h = 9

lower part of the wall is 9 meters

using the sine ratio in the right triangle on the right

sin60° =
`(opposite)/(hypotenuse)` =
`(9+x)/(18)` =
`(√(3) )/(2)` ( cross- multiply )

2(9 + x) = 18
`√(3)` ( divide both sides by 2 )

9 + x = 9
`√(3)` ( subtract 9 from both sides )

x = 9
`√(3)` - 9 = 9(
`√(3)` - 1)

Answer 2

`\textsf{A)} \quad x=9(√(3)-1)`

Explanation:

Interior angles of a triangle sum to 180°. Therefore, if the ladder makes an angle of 30° with the ground, the angle it makes with the wall is 60°.

Therefore, the two triangles are congruent as their corresponding angles are the same and the length of their longest side (hypotenuse) is the same.

30-60-90 Triangle Theorem

A 30-60-90 triangle is a special right triangle where the measures of its angles are in the ratio 1 : 2 : 3 and the measure of its sides are in the ratio 1 : √3 : 2.

The formula for the sides is b : b√3 : 2b where:

• b = the side opposite the 30° angle (shortest leg).
• b√3 = the side opposite the 60° angle.
• 2b = the side opposite the right angle (hypotenuse).

Therefore, as the length of the hypotenuse is 18:

`\implies 2b=18`

`\implies b=9`

`\implies b√(3)=9√(3)`

To find x, subtract the length of the shortest leg of the right triangle from the length of the other leg of the right triangle:

`\implies x=9√(3)-9`

`\implies x=9(√(3)-1)`