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A flower bed is in the shape of a triangle with one side twice the length of the shortest side, and the third side 15 feet longer than the shortest side. If the perimeter is 100 feet and [tex]$x$[/tex] represents the length of the shortest side, find an equation to solve for the lengths of the three sides.

A. [tex]$x + 2x + x + 15 = 100$[/tex]
B. [tex]$x + 15 = 2x$[/tex]
C. [tex]$x + 15 + 2x = 100$[/tex]
D. [tex]$x + 2x = x + 115$[/tex]

Sagot :

GinnyAnswer
Certainly! Let's go through the problem step-by-step to understand the solution.

### Step 1: Define Variables
Let \( x \) represent the length of the shortest side of the flower-bed.

### Step 2: Express the Other Sides in Terms of \( x \)
- One side of the triangle is twice the length of the shortest side. Therefore, the length of this side is \( 2x \).
- The third side is 15 feet longer than the shortest side. Therefore, the length of this side is \( x + 15 \).

### Step 3: Set Up the Perimeter Equation
The perimeter of a triangle is the sum of the lengths of all three sides. According to the problem, the perimeter is 100 feet. Therefore, we can write the equation as:
[tex]\[ x + 2x + (x + 15) = 100 \][/tex]

### Step 4: Simplify the Equation
Combine like terms to simplify the equation:
[tex]\[ x + 2x + x + 15 = 100 \][/tex]
[tex]\[ 4x + 15 = 100 \][/tex]

### Step 5: Solve for \( x \)
Isolate \( x \) by performing the following steps:
1. Subtract 15 from both sides:
[tex]\[ 4x + 15 - 15 = 100 - 15 \][/tex]
[tex]\[ 4x = 85 \][/tex]
2. Divide both sides by 4:
[tex]\[ x = \frac{85}{4} \][/tex]
[tex]\[ x = 21.25 \][/tex]

So, the length of the shortest side is \( 21.25 \) feet.

### Step 6: Find the Lengths of the Other Sides
Now, we can determine the lengths of the other sides:
- The second side is \( 2x \):
[tex]\[ 2 \cdot 21.25 = 42.5 \][/tex] feet
- The third side is \( x + 15 \):
[tex]\[ 21.25 + 15 = 36.25 \][/tex] feet

### Conclusion
The lengths of the three sides of the flower-bed triangle are:
- Shortest side: \( 21.25 \) feet
- Second side: \( 42.5 \) feet
- Third side: \( 36.25 \) feet

### Answering the Equation Question
The correct equation that we used to set up the problem is:
[tex]\[ x + 2x + (x + 15) = 100 \][/tex]

Thus, the correct answer is:
[tex]\[ A) \, x + 2x + x + 15 = 100 \][/tex]
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