Answer:
A
Step-by-step explanation:
using the sine ratio and the exact values
sin30° = [tex]\frac{1}{2}[/tex] and sin60° = [tex]\frac{\sqrt{3} }{2}[/tex]
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° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{h}{18}[/tex] = [tex]\frac{1}{2}[/tex] ( 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° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{9+x}{18}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
2(9 + x) = 18[tex]\sqrt{3}[/tex] ( divide both sides by 2 )
9 + x = 9[tex]\sqrt{3}[/tex] ( subtract 9 from both sides )
x = 9[tex]\sqrt{3}[/tex] - 9 = 9([tex]\sqrt{3}[/tex] - 1)
Answer:
[tex]\textsf{A)} \quad x=9(\sqrt{3}-1)[/tex]
Step-by-step 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:
Therefore, as the length of the hypotenuse is 18:
[tex]\implies 2b=18[/tex]
[tex]\implies b=9[/tex]
[tex]\implies b\sqrt{3}=9\sqrt{3}[/tex]
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:
[tex]\implies x=9\sqrt{3}-9[/tex]
[tex]\implies x=9(\sqrt{3}-1)[/tex]
