Q. 443 - Intermediate Algebra

Question

Solve:

A ladder leans against the wall of a building. The length of the ladder is $9$feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is $7$feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

Step-by-Step Solution

Answer

The length of all three sides of the triangle formed by the ladder leaning against the building is $8,15,$and $17$feet

Step 1. Given the information.

The length of the ladder is $9$ feet longer than the distance of the bottom of the ladder from the building.

 The distance of the top of the ladder reaches up the side of the building is $7$feet longer than the distance of the bottom of the ladder from the building.

Step 2. Pythagorean formula.

To find the length of all three sides of the triangle formed by the ladder leaning against the building we use the formula of Pythagorean Theorem,

that is, $a^2+b^2=c^2$.

Step 3. Assuming the sides of the triangle.

Let the distance of the bottom of the ladder from the building be $x$feet.

The length of the ladder is, $(x+9)$feet

Distance of the top of the ladder reaches the side of the building$=(x+7)$feet.

Step 4. Finding the three sides.

The Pythagorean formula is,

$a^2+b^2=c^2$

Inputting the values,

$x^2+{(x+7)}^2={(x+9)}^2$

Using the identity ${(a+b)}^2=a^2+2ab+b^2$,

$$\begin{gathered} x^2+x^2+14x+49=x^2+18x+81 \\ 2x^2+14x+49=x^2+18x+81 \\ 2x^2-x^2+14x-18x=81-49 \\ {x}^2-4x=32 \\ {x}^2-4x-32=0 \end{gathered}$$

Splitting the middle term such that the sum of the terms is $(-4)$ and the product of the terms is $(-32)$.

$$\begin{gathered} x^2-8x+4x-32=0 \\ (x^2-8x)+(4x-32)=0 \end{gathered}$$

Taking out the common factors,

$$\begin{gathered} x(x-8)+4(x-8)=0 \\ (x-8)(x+4)=0 \end{gathered}$$

$x=8$

or, $x=-4$

Since the length of a side cannot be negative.

Length of the side is, $x=8$.

Distance of the bottom of the ladder from the building is,

$x=8feet$

Length of the ladder is,

$$\begin{gathered} x+9 \\ =8+9 \\ =17feet \end{gathered}$$

Distance of the top of the ladder reaches the building is,

$$\begin{gathered} x+7 \\ =8+7 \\ =15feet \end{gathered}$$

Step 5. Checking the solution

The Pythagorean theorem is,

$a^2+b^2=c^2$

Substituting the values and checking,

$$\begin{gathered} 8^2+{15}^2={17}^2 \\ 64+225=289 \\ 289=289 \end{gathered}$$

This is true.