Find solutions to your problems with the help of IDNLearn.com's expert community. Find accurate and detailed answers to your questions from our experienced and dedicated community members.
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]
### 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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.