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The perimeter of a rectangle, whose adjacent sides are represented by [tex]3x[/tex] and [tex]2x+3[/tex], is equal to the perimeter of an equilateral triangle whose side is represented by [tex]5x-3[/tex]. Find the value of [tex]x[/tex].

[tex]x = \square[/tex]


Sagot :

To find the value of [tex]\( x \)[/tex], we need to set up equations for the perimeters of the rectangle and the equilateral triangle, and then equate them.

### Step-by-Step Solution

1. Perimeter of the Rectangle:

The rectangle has sides [tex]\(3x\)[/tex] and [tex]\(2x + 3\)[/tex].

The perimeter [tex]\(P_{\text{rectangle}}\)[/tex] of a rectangle is given by:
[tex]\[ P_{\text{rectangle}} = 2 \times (\text{length} + \text{width}) \][/tex]

Here, the length is [tex]\(3x\)[/tex] and the width is [tex]\(2x + 3\)[/tex]. Thus,
[tex]\[ P_{\text{rectangle}} = 2 \times (3x + (2x + 3)) = 2 \times (3x + 2x + 3) = 2 \times (5x + 3) \][/tex]

Simplifying further,
[tex]\[ P_{\text{rectangle}} = 10x + 6 \][/tex]

2. Perimeter of the Equilateral Triangle:

The side of the equilateral triangle is [tex]\(5x - 3\)[/tex].

The perimeter [tex]\(P_{\text{triangle}}\)[/tex] of an equilateral triangle is given by:
[tex]\[ P_{\text{triangle}} = 3 \times (\text{side}) \][/tex]

Here, the side is [tex]\(5x - 3\)[/tex]. Thus,
[tex]\[ P_{\text{triangle}} = 3 \times (5x - 3) \][/tex]

Simplifying further,
[tex]\[ P_{\text{triangle}} = 15x - 9 \][/tex]

3. Set Perimeters Equal to Each Other:

Since the perimeter of the rectangle is equal to the perimeter of the equilateral triangle, we set the two expressions equal to each other:
[tex]\[ 10x + 6 = 15x - 9 \][/tex]

4. Solve for [tex]\(x\)[/tex]:

Rearrange the equation to isolate [tex]\(x\)[/tex]:
[tex]\[ 10x + 6 = 15x - 9 \][/tex]

Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[ 6 = 5x - 9 \][/tex]

Add 9 to both sides:
[tex]\[ 15 = 5x \][/tex]

Divide both sides by 5:
[tex]\[ x = 3 \][/tex]

Hence, the value of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]