To solve this problem, we need to find the perimeter of a triangle when its sides are in the extended ratio of [tex]\(3:4:10\)[/tex] and the shortest side is given as 9 inches.
Let's break it down step-by-step:
1. Identify Given Information:
- The side ratios of the triangle are [tex]\(3:4:10\)[/tex].
- The shortest side of the triangle, which corresponds to the first ratio value, is 9 inches.
2. Express Sides in Terms of a Common Multiplier:
Since the sides are in the ratio [tex]\(3:4:10\)[/tex], we can express the sides of the triangle as:
[tex]\[
3x, \; 4x, \; 10x
\][/tex]
where [tex]\(x\)[/tex] is the common multiplier.
3. Find the Value of [tex]\(x\)[/tex]:
The shortest side is given as 9 inches, which corresponds to [tex]\(3x\)[/tex]. Therefore, we can solve for [tex]\(x\)[/tex]:
[tex]\[
3x = 9
\][/tex]
Dividing both sides by 3, we get:
[tex]\[
x = 3
]
4. Calculate the Lengths of the Other Sides:
Using the value of \(x\), we can find the lengths of the other sides:
\[
\text{First side} = 3x = 3 \cdot 3 = 9 \text{ inches}
\][/tex]
[tex]\[
\text{Second side} = 4x = 4 \cdot 3 = 12 \text{ inches}
\][/tex]
[tex]\[
\text{Third side} = 10x = 10 \cdot 3 = 30 \text{ inches}
\][/tex]
5. Calculate the Perimeter of the Triangle:
The perimeter of the triangle is the sum of all its sides:
[tex]\[
\text{Perimeter} = 9 + 12 + 30 = 51 \text{ inches}
\][/tex]
Therefore, the perimeter of the triangle is [tex]\( 51 \)[/tex] inches.
