Sure! Let's solve this step-by-step.
1. Understanding the Problem:
- The sum of the diagonals of the rectangle is given as 26 cm.
- One of the sides of the rectangle is given as 5 cm.
- We need to find the sum of the lengths of the other three sides of the rectangle.
2. Find the Length of one diagonal:
Since the sum of the diagonals is 26 cm and the diagonals of a rectangle are equal in length, each diagonal would be:
[tex]\[
\text{Length of one diagonal} = \frac{26}{2} = 13 \text{ cm}
\][/tex]
3. Using the Pythagorean theorem:
The diagonals of a rectangle divide it into two right-angled triangles. Using the Pythagorean theorem in one of these triangles, we can find the length of the other side. Let's denote the unknown side as [tex]\( b \)[/tex].
The relationship is given by:
[tex]\[
\text{Diagonal}^2 = \text{side}_a^2 + \text{side}_b^2
\][/tex]
Substituting the known values:
[tex]\[
13^2 = 5^2 + b^2
\][/tex]
[tex]\[
169 = 25 + b^2
\][/tex]
Now, solving for [tex]\( b \)[/tex]:
[tex]\[
b^2 = 169 - 25 = 144
\][/tex]
[tex]\[
b = \sqrt{144} = 12 \text{ cm}
\][/tex]
4. Calculate the sum of the lengths of the other three sides:
- Sum of the other three sides of the rectangle includes:
- The given side [tex]\( a = 5 \text{ cm} \)[/tex]
- The unknown side [tex]\( b = 12 \text{ cm} \)[/tex]
- Another instance of side [tex]\( a = 5 \text{ cm} \)[/tex] and side [tex]\( b = 12 \text{ cm} \)[/tex]
So, the sum is:
[tex]\[
\text{Sum} = \text{side}_a + \text{side}_b + \text{side}_b = 5 + 5 + 12 + 12 = 34 \text{ cm}
\][/tex]
Therefore, the sum of the lengths of the other three sides of the rectangle is 34 cm.
