Question

the perimeter of the triangular banner is 13 feet. what is the length of the shortest side?
$(x - 5)$ in.
$\frac{x}{2}$ in.
6 in.
\bigcirc 3 ft
\bigcirc 4 ft
\bigcirc 6 ft
\bigcirc 8 ft

Explanation:

Step1: Convert perimeter to inches

Since 1 foot = 12 inches, $13 \times 12 = 156$ inches.

Step2: Set up perimeter equation

Sum of sides equals total perimeter:
$$(x-5) + \frac{x}{2} + 6 = 156$$

Step3: Simplify the equation

Combine like terms and solve for $x$:
$$x + \frac{x}{2} + 1 = 156$$
$$\frac{3x}{2} = 155$$
$$3x = 310$$
$$x = \frac{310}{3} \approx 103.33$$

Step4: Calculate each side length

• Side 1: $x-5 = \frac{310}{3} - 5 = \frac{295}{3} \approx 98.33$ inches, or $\frac{295}{36} \approx 8.2$ feet
• Side 2: $\frac{x}{2} = \frac{310}{6} \approx 51.67$ inches, or $\frac{155}{36} \approx 4.3$ feet
• Side 3: 6 inches = $\frac{6}{12} = 0.5$ feet

*Note: There is a discrepancy between the answer choices and the calculated value, but rechecking the perimeter conversion and equation confirms the math. However, if we assume the perimeter was mistakenly given in inches (13 inches instead of 13 feet):
Revised Step1: Use perimeter = 13 inches
Revised Step2: $(x-5) + \frac{x}{2} + 6 = 13$
Revised Step3: $\frac{3x}{2} +1 =13 \implies \frac{3x}{2}=12 \implies x=8$
Revised Step4:

• Side1: $8-5=3$ inches = $\frac{3}{12}=0.25$ ft (not an option)
• Side2: $\frac{8}{2}=4$ inches = $\frac{4}{12}\approx0.33$ ft (not an option)
• Side3: 6 inches = 0.5 ft (not an option)

Assuming the question intended the perimeter in feet but side units in feet (not inches):
Revised Step1: Perimeter =13 ft
Revised Step2: $(x-5) + \frac{x}{2} +6=13$
Revised Step3: $\frac{3x}{2}+1=13 \implies x=8$
Revised Step4:

• Side1: $8-5=3$ ft
• Side2: $\frac{8}{2}=4$ ft
• Side3: 6 ft

This matches the answer choices, so we use this corrected unit assumption.*

Step5: Identify the shortest side

Compare the three side lengths: 3 ft, 4 ft, 6 ft.

Answer:

A. 3 ft