To solve the problem of finding the ratio of the sides of a rectangle where the diagonal is thrice the smaller side, we'll follow these steps:
1. Define Variables:
- Let the smaller side of the rectangle be [tex]\( a \)[/tex].
- Let the larger side of the rectangle be [tex]\( b \)[/tex].
2. Apply the Pythagorean Theorem:
- For a rectangle, the diagonal forms a right triangle with the sides.
- So, the length of the diagonal can be expressed using Pythagoras' theorem as:
[tex]\[
\sqrt{a^2 + b^2}
\][/tex]
3. Given Condition:
- The diagonal is thrice the smaller side, meaning:
[tex]\[
\text{Diagonal} = 3a
\][/tex]
- Hence, we have the equation:
[tex]\[
3a = \sqrt{a^2 + b^2}
\][/tex]
4. Solve for [tex]\( b \)[/tex]:
- Square both sides of the equation to eliminate the square root:
[tex]\[
(3a)^2 = a^2 + b^2
\][/tex]
- Simplify the equation:
[tex]\[
9a^2 = a^2 + b^2
\][/tex]
- Subtract [tex]\( a^2 \)[/tex] from both sides:
[tex]\[
8a^2 = b^2
\][/tex]
- Take the square root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \sqrt{8a^2} = \sqrt{8} \cdot a = 2\sqrt{2} \cdot a
\][/tex]
5. Determine the Ratio:
- The ratio of the larger side to the smaller side [tex]\( \frac{b}{a} \)[/tex] is:
[tex]\[
\frac{b}{a} = \frac{2\sqrt{2} \cdot a}{a} = 2\sqrt{2}
\][/tex]
Thus, the ratio of the sides of the rectangle is [tex]\( 2\sqrt{2} \)[/tex].
