Question

Find the lengths of the sides of the rectangle if it is known that one of them is 14cm bigger than the other, and the diagonal of the rectangle is 34Cm

Answer 1

Let's call the smaller side $s$. This means that the other side of the rectangle is $s + 14$.

In this problem, we are trying to find the lengths of the sides. The problem gives us someinformation about the sides, but that's really it. However, the problem also let's us know that the diagonal of the rectangle (the line connecting opposite corners of the rectangle) is 34 cm long.

This fact is very important, because we can actually make a triangle, with the smaller side, bigger side, and the diagonal of the rectangle. Additionally, since we are working with rectangles, we know that the sides form a right angle. Since we are constructing a triangle with the sides as the legs, we know that we are going to be constructing a right triangle, which means that we can work with Pythagorean's Theorem.

Remember that Pythaogrean's Theorem is:

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

• $a$ and $b$ are the lengths of the legs of the triangle
• $c$ is the length of the hypotenuse

Applying the Pythagorean Theorem to this problem, we get:

$s^2 + (s + 14)^2 = 34^2$

Let's simplify and solve for $s$:

$s^2 + (s + 14)^2 = 34^2$

• Set up

$s^2 + (s^2 + 28s + 196) = 1156$

• Simplify left hand side and evalutate $34^2$

$2s^2 + 28s -962 = 0$

• Subtract 1156 from both sides of the equation and combine like terms

$(s + 30)(s - 16) = 0$

• Factor

$s = -30, 16$

• Apply the Zero Product Property to solve for $s$

$s = 16$

• $s = -30$ is an extraneous solution because you can't have a negative side length

We have now found that the smaller side is 16 cm long. Since the larger side is 14 cm longer, it can be found as shown:

$16 \,\textrm{cm} + 14 \,\textrm{cm} = 30 \,\textrm{cm}$

The sides are length 16 cm and 30 cm.