# A ladder of $3.9m$ length is laid against a wall. The distance between the foot of the wall and the ladder is $1.5m$. Find the height at which the ladder touches the wall.

(A) $3.6m$ (B) $4.55m$ (C) $4.82m$ (D) $5.7m$

Last updated date: 19th Mar 2023

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Answer

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Hint: The ladder, the wall and the ground will form a right angled triangle. Apply Pythagoras theorem to find the height.

According to the question, the length of the ladder is $3.9m$.

When the ladder leans against the wall, it will form a right angled triangle with ladder as the hypotenuse, height of the point of contact of ladder to the wall from the ground as perpendicular and the distance between the foot of wall and the point of contact of ladder with the ground as base of the triangle. So, we have:

Hypotenuse,\[H = 3.9m\] and Base\[B = 1.5m\].

Applying Pythagoras theorem to find perpendicular:

\[

\Rightarrow {\left( H \right)^2} = {\left( B \right)^2} + {\left( P \right)^2}, \\

\Rightarrow {P^2} = {H^2} - {B^2}, \\

\Rightarrow {P^2} = {\left( {3.9} \right)^2} - {\left( {1.5} \right)^2}, \\

\Rightarrow P = \sqrt {15.21 - 2.25} , \\

\Rightarrow P = \sqrt {12.96} , \\

\Rightarrow P = 3.6m \\

\]

Thus, the ladder touches the wall $3.6m$ above the ground.

Note: If in case the angle made by the ladder with the wall or with the ground is known, we can also use trigonometric ratios to solve such types of questions.

According to the question, the length of the ladder is $3.9m$.

When the ladder leans against the wall, it will form a right angled triangle with ladder as the hypotenuse, height of the point of contact of ladder to the wall from the ground as perpendicular and the distance between the foot of wall and the point of contact of ladder with the ground as base of the triangle. So, we have:

Hypotenuse,\[H = 3.9m\] and Base\[B = 1.5m\].

Applying Pythagoras theorem to find perpendicular:

\[

\Rightarrow {\left( H \right)^2} = {\left( B \right)^2} + {\left( P \right)^2}, \\

\Rightarrow {P^2} = {H^2} - {B^2}, \\

\Rightarrow {P^2} = {\left( {3.9} \right)^2} - {\left( {1.5} \right)^2}, \\

\Rightarrow P = \sqrt {15.21 - 2.25} , \\

\Rightarrow P = \sqrt {12.96} , \\

\Rightarrow P = 3.6m \\

\]

Thus, the ladder touches the wall $3.6m$ above the ground.

Note: If in case the angle made by the ladder with the wall or with the ground is known, we can also use trigonometric ratios to solve such types of questions.

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