Question

Is the length of a side of square $S$ greater than the length of a side of equilateral triangle $T?$ (1) The sum of the lengths of a side of $S$ and a side of $T$ is 22 . (2) The ratio of the perimeter of square $S$ to the perimeter of triangle $T$ is 5 to 6

Short answer

The length of a side of square $S$ is not greater than the length of a side of triangle $T$.

Step-by-step solution

Understand the Problem

To determine whether the length of a side of square $S$ is greater than the length of a side of equilateral triangle $T$, we need to use the given conditions: (1) the sum of the lengths of a side of $S$ and a side of $T$ is 22, and (2) the ratio of the perimeter of square $S$ to the perimeter of triangle $T$ is 5 to 6.

Define Variables

Let $s$ be the length of a side of square $S$ and $t$ be the length of a side of equilateral triangle $T$.

Use Condition (1)

From condition (1), we have the equation: $$s+t=22$$

Use Condition (2)

Determine the perimeters of the shapes. The perimeter of square $S$ is $4s$, and the perimeter of triangle $T$ is $3t$. According to condition (2): $$\frac{4s}{3t}=\frac{5}{6}$$

Solve the Ratio Equation for $s$

From condition (2): $$4s=\frac{5}{6}\times 3t$$ Simplify: $$4s=\frac{5}{2}t$$ Solve for $s$: $$s=\frac{5}{8}t$$

Substitute into the Sum Equation

Substitute $s$ into the equation from condition (1): $$\frac{5}{8}t+t=22$$ Combine like terms: $$\frac{5}{8}t+\frac{8}{8}t=22$$ $$\frac{13}{8}t=22$$

Solve for $t$

Multiply both sides by 8: $$13t=176$$ Divide by 13: $$t=\frac{176}{13}=13.54$$ (approx)

Find $s$

Substitute $t$ back into the equation for $s$: $$s=\frac{5}{8}\times 13.54$$ $$s=8.46$$ (approx)

Compare $s$ and $t$

$s=8.46$ and $t=13.54$, so the length of a side of square $S$ is less than the length of a side of triangle $T$.

Key concepts

Square Side Length Comparison

In this section, we explore how to compare the side lengths of different geometric shapes, specifically a square and an equilateral triangle. To determine if one length is greater than the other, we need specific information about their lengths. We are given two conditions:
1. The sum of the lengths of a side of the square (denoted as $s$) and a side of the equilateral triangle (denoted as $t$) is 22.
2. The ratio of the perimeter of the square to the perimeter of the triangle is 5 to 6.

By using these conditions, we can set up equations to solve for the side lengths of the square and the triangle. This will allow us to directly compare their lengths.

Equilateral Triangle Side Length

An equilateral triangle has three sides of equal length. When we say the side length of an equilateral triangle is denoted by $t$, it means each side of this triangle is $t$ units long. The perimeter of an equilateral triangle is simply three times the length of one side, so the perimeter is $3t$.

Given in our problem, we can use the relationship between the perimeters and the given ratio to find $t$. Let's see an example:

The ratio of the perimeter of the square ($4s$) to the perimeter of the triangle ($3t$) is 5 to 6, represented mathematically as:
$$\frac{4s}{3t}=\frac{5}{6}$$
From this equation, we can solve for one variable in terms of the other. In our solution, we found that $s=\frac{5}{8}t$. This relationship helps in finding the precise lengths when combined with the other given condition.

Perimeter Ratios

Perimeter ratios are crucial in comparing geometric shapes. They help us understand how the perimeter of one shape relates to the perimeter of another. In this problem, the ratio given is 5 to 6 for the perimeters of a square and an equilateral triangle, respectively. This ratio provides a way to derive the necessary equations and find the side lengths of each shape.

We know the perimeter of the square is ($4s$) and the perimeter of the triangle is ($3t$). Setting up the ratio, we have:
$$\frac{4s}{3t}=\frac{5}{6}$$
Simplifying this, we get:
$$4s=\frac{5}{2}t$$
By solving this, we can substitute $s$ into our sum equation from condition (1):
$$\frac{5}{8}t+t=22$$
This converts the given problem into an algebraic equation that can be solved to find the lengths $s$ and $t$.

Through these calculations, we discovered that $s$ (the side of the square) is approximately 8.46, and $t$ (the side of the triangle) is approximately 13.54. Thus, the side length of the square is less than that of the equilateral triangle.