To determine the side lengths of a right triangle where the hypotenuse is 5 inches and the longer leg is 1 inch longer than the shorter leg, we can follow these steps:
1. Identify Given Information:
- The hypotenuse (c) is 5 inches.
- Let [tex]\( x \)[/tex] be the length of the shorter leg.
- The longer leg is [tex]\( x + 1 \)[/tex] inches.
2. Use the Pythagorean Theorem:
According to the Pythagorean Theorem for a right triangle:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Here, [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the legs, and [tex]\( c \)[/tex] is the hypotenuse.
3. Formulate the Equation:
Substituting the values, we get:
[tex]\[
x^2 + (x + 1)^2 = 5^2
\][/tex]
4. Simplify the Equation:
[tex]\[
x^2 + (x + 1)^2 = 25
\][/tex]
Expand the squared term:
[tex]\[
x^2 + (x^2 + 2x + 1) = 25
\][/tex]
Combine like terms:
[tex]\[
2x^2 + 2x + 1 = 25
\][/tex]
5. Set Up the Quadratic Equation:
[tex]\[
2x^2 + 2x + 1 - 25 = 0
\][/tex]
Simplify further:
[tex]\[
2x^2 + 2x - 24 = 0
\][/tex]
Divide the entire equation by 2:
[tex]\[
x^2 + x - 12 = 0
\][/tex]
6. Solve the Quadratic Equation:
Using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -12 \)[/tex]:
[tex]\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-1 \pm \sqrt{1 + 48}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{49}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm 7}{2}
\][/tex]
Therefore, we get two solutions:
[tex]\[
x = \frac{6}{2} = 3
\][/tex]
[tex]\[
x = \frac{-8}{2} = -4
\][/tex]
We discard the negative solution since a length cannot be negative.
7. Determine the Side Lengths:
- The shorter leg [tex]\( x = 3 \)[/tex] inches.
- The longer leg [tex]\( x + 1 = 4 \)[/tex] inches.
8. Conclusion:
- Length of the shorter leg: [tex]\( 3 \)[/tex] inches.
- Length of the longer leg: [tex]\( 4 \)[/tex] inches.
- Length of the hypotenuse: [tex]\( 5 \)[/tex] inches.
These findings confirm that the side lengths of the triangle are 3 inches, 4 inches, and 5 inches, respectively.
