Question

Find the value of x so that the figures have the same perimeter.

A 4 B 5 C 6 D 7

Short answer

The value of x so that the figures have the same perimeter is 4. So, the correct option is A.

Step-by-step solution

Step 1. State the formula.

The perimeter of a triangle is the sum of the lengths of its sides. The perimeter of a rectangle is given as $2\left(l+w\right)$, where $l$is the length of the rectangle and $w$is the width of the rectangle.

Step 2. List the given data.

From the figure, the lengths of the sides of the triangle are $3x+4$, $5x+1$and $2x+5$units respectively. Then, the perimeter of the first figure is $3x+4+5x+1+2x+5$units.

Similarly, from the given figure, the length of the rectangle is $x+13$units and the width of the second rectangle is $2x$units. Then, the perimeter of the second rectangle is $2\left(x+13+2x\right)$units.

Step 3. Formulate the equation.

It is given that the figures have the same perimeter. Then, the perimeters of the triangle and the rectangle are equal. So, “$3x+4+5x+1+2x+5=2\left(x+13+2x\right)$”.

This is the required equation.

Step 4. Solve the obtained equation.

Solving,

$3x+4+5x+1+2x+5=2\left(x+13+2x\right)$ (Given equation)

$10x+10=2\left(3x+13\right)$ (Simplify)

$10x+10=6x+26$ (Distributive property)

$10x+10-6x=6x+26-6x$ (Subtract $6x$from both sides)

$4x+10=26$ (Simplify)

$4x+10-10=26-10$ (Subtract 10 from both sides)

$4x=16$ (Simplify)

$\frac{4x}{4}=\frac{16}{4}$ (Divide both sides by 4)

$x=4$ (Simplify)

So, $x=4$is the solution of the given equation.

Step 5. Interpret the obtained solution.

It was assumed that the perimeters of both figures are equal. This assumption led to the obtained solution $x=4$.

Thus, the value of x so that the figures have the same perimeter is 4.