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Answer :
To solve this problem, we want to find an equation that helps us determine the value of [tex]\( x \)[/tex] in an isosceles triangle with a perimeter of 7.5 meters, where the shortest side, [tex]\( y \)[/tex], measures 2.1 meters.
### Step-by-Step Solution:
1. Identify the characteristics of the isosceles triangle:
- An isosceles triangle has two equal sides. Let the equal sides each be [tex]\( x \)[/tex].
- The shortest side is given as [tex]\( y = 2.1 \)[/tex] meters.
2. Understand the relationship of sides for the perimeter:
- The formula for the perimeter [tex]\( P \)[/tex] of any triangle is:
[tex]\[
P = \text{sum of the lengths of all sides}
\][/tex]
- Therefore, for the isosceles triangle:
[tex]\[
P = 2x + y
\][/tex]
- Plug in the known values: [tex]\( P = 7.5 \)[/tex] meters and [tex]\( y = 2.1 \)[/tex] meters:
[tex]\[
7.5 = 2x + 2.1
\][/tex]
3. Solve for the correct equation:
- Rearrange the equation to reflect the relationship between [tex]\( x \)[/tex] and the known values:
[tex]\[
2x + 2.1 = 7.5
\][/tex]
4. Check the given equation options:
From the listed options:
- [tex]\( 2x - 2.1 = 7.5 \)[/tex]
- [tex]\( 4.2 + y = 7.5 \)[/tex]
- [tex]\( v - 4.2 = 7.5 \)[/tex]
- [tex]\( 2.1 + 2x = 7.5 \)[/tex]
Checking each option:
- Option 1: [tex]\( 2x - 2.1 = 7.5 \)[/tex] doesn't relate to our derived equation correctly.
- Option 2: [tex]\( 4.2 + y = 7.5 \)[/tex] is incorrect as it doesn't involve [tex]\( x \)[/tex].
- Option 3: [tex]\( v - 4.2 = 7.5 \)[/tex] replaces [tex]\( x \)[/tex] incorrectly and has a different approach.
- Option 4: [tex]\( 2.1 + 2x = 7.5 \)[/tex] matches our derived equation and therefore is the correct one.
Thus, the correct equation to use is:
[tex]\[ 2.1 + 2x = 7.5 \][/tex]
This equation properly reflects the relationship of the sides in the isosceles triangle knowing its perimeter and the value of the shortest side.
### Step-by-Step Solution:
1. Identify the characteristics of the isosceles triangle:
- An isosceles triangle has two equal sides. Let the equal sides each be [tex]\( x \)[/tex].
- The shortest side is given as [tex]\( y = 2.1 \)[/tex] meters.
2. Understand the relationship of sides for the perimeter:
- The formula for the perimeter [tex]\( P \)[/tex] of any triangle is:
[tex]\[
P = \text{sum of the lengths of all sides}
\][/tex]
- Therefore, for the isosceles triangle:
[tex]\[
P = 2x + y
\][/tex]
- Plug in the known values: [tex]\( P = 7.5 \)[/tex] meters and [tex]\( y = 2.1 \)[/tex] meters:
[tex]\[
7.5 = 2x + 2.1
\][/tex]
3. Solve for the correct equation:
- Rearrange the equation to reflect the relationship between [tex]\( x \)[/tex] and the known values:
[tex]\[
2x + 2.1 = 7.5
\][/tex]
4. Check the given equation options:
From the listed options:
- [tex]\( 2x - 2.1 = 7.5 \)[/tex]
- [tex]\( 4.2 + y = 7.5 \)[/tex]
- [tex]\( v - 4.2 = 7.5 \)[/tex]
- [tex]\( 2.1 + 2x = 7.5 \)[/tex]
Checking each option:
- Option 1: [tex]\( 2x - 2.1 = 7.5 \)[/tex] doesn't relate to our derived equation correctly.
- Option 2: [tex]\( 4.2 + y = 7.5 \)[/tex] is incorrect as it doesn't involve [tex]\( x \)[/tex].
- Option 3: [tex]\( v - 4.2 = 7.5 \)[/tex] replaces [tex]\( x \)[/tex] incorrectly and has a different approach.
- Option 4: [tex]\( 2.1 + 2x = 7.5 \)[/tex] matches our derived equation and therefore is the correct one.
Thus, the correct equation to use is:
[tex]\[ 2.1 + 2x = 7.5 \][/tex]
This equation properly reflects the relationship of the sides in the isosceles triangle knowing its perimeter and the value of the shortest side.
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