Question
The area of the right triangle shown is 24 square feet.
Which equations can be used to find the lengths of the legs of the triangle? Select three options.
0.5(x)(x + 2) = 24
x(x + 2) = 24
x2 + 2x – 24 = 0
x2 + 2x – 48 = 0
x2 + (x + 2)2 = 100
Which equations can be used to find the lengths of the legs of the triangle? Select three options.
0.5(x)(x + 2) = 24
x(x + 2) = 24
x2 + 2x – 24 = 0
x2 + 2x – 48 = 0
x2 + (x + 2)2 = 100
Answers
GPT-4o mini
To find the lengths of the legs of a right triangle given that its area is 24 square feet, we can use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Let’s assume one leg of the triangle is \(x\) and the other leg is \(x + 2\). Therefore, the area can be expressed as:
\[
\frac{1}{2} \times x \times (x + 2) = 24
\]
To eliminate the fraction, we can multiply both sides by 2:
\[
x(x + 2) = 48
\]
This equation can be used to find the value of \(x\). So, this option is correct:
- **\(x(x + 2) = 48\)**
Now let’s also consider what the options entail:
1. **\(0.5(x)(x + 2) = 24\)**: This equation is equivalent to the area formula we derived above and is also correct.
2. **\(x^2 + 2x - 24 = 0\)**: This is not correct because it does not represent the area of the triangle—this would imply that the area is \(12\) instead of \(24\).
3. **\(x^2 + 2x - 48 = 0\)**: This equation is derived from setting \(x(x + 2) = 48\) to zero. It can be used to find the lengths of the legs, so this is also correct.
4. **\(x^2 + (x + 2)^2 = 100\)**: This represents the Pythagorean theorem but does not relate to area calculation, so it's not directly useful for finding the legs based on the area provided.
To summarize, the three valid options to solve for the triangle legs are:
1. **\(0.5(x)(x + 2) = 24\)**
2. **\(x(x + 2) = 48\)**
3. **\(x^2 + 2x - 48 = 0\)**
So, the correct answers are:
- **0.5(x)(x + 2) = 24**
- **x(x + 2) = 48**
- **x^2 + 2x - 48 = 0**
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Let’s assume one leg of the triangle is \(x\) and the other leg is \(x + 2\). Therefore, the area can be expressed as:
\[
\frac{1}{2} \times x \times (x + 2) = 24
\]
To eliminate the fraction, we can multiply both sides by 2:
\[
x(x + 2) = 48
\]
This equation can be used to find the value of \(x\). So, this option is correct:
- **\(x(x + 2) = 48\)**
Now let’s also consider what the options entail:
1. **\(0.5(x)(x + 2) = 24\)**: This equation is equivalent to the area formula we derived above and is also correct.
2. **\(x^2 + 2x - 24 = 0\)**: This is not correct because it does not represent the area of the triangle—this would imply that the area is \(12\) instead of \(24\).
3. **\(x^2 + 2x - 48 = 0\)**: This equation is derived from setting \(x(x + 2) = 48\) to zero. It can be used to find the lengths of the legs, so this is also correct.
4. **\(x^2 + (x + 2)^2 = 100\)**: This represents the Pythagorean theorem but does not relate to area calculation, so it's not directly useful for finding the legs based on the area provided.
To summarize, the three valid options to solve for the triangle legs are:
1. **\(0.5(x)(x + 2) = 24\)**
2. **\(x(x + 2) = 48\)**
3. **\(x^2 + 2x - 48 = 0\)**
So, the correct answers are:
- **0.5(x)(x + 2) = 24**
- **x(x + 2) = 48**
- **x^2 + 2x - 48 = 0**