The hypotenuse of a right triangle is
13 inches long. The shorter leg is
7 inches shorter than the longer leg. Find the side lengths of the triangle.
5, 12, 13
To find the side lengths of the right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's denote the longer leg as 'x' inches.
According to the problem, the shorter leg is 7 inches shorter than the longer leg. So, the length of the shorter leg can be represented as 'x - 7' inches.
Now, we can apply the Pythagorean theorem:
(hypotenuse)^2 = (longer leg)^2 + (shorter leg)^2
Substituting the given values:
13^2 = x^2 + (x - 7)^2
Expanding the equation:
169 = x^2 + (x^2 - 14x + 49)
Combining like terms:
169 = 2x^2 - 14x + 49
Rearranging the equation to set it equal to zero:
0 = 2x^2 - 14x + 49 - 169
0 = 2x^2 - 14x - 120
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, it can be factored as:
0 = 2(x^2 - 7x - 60)
0 = 2(x - 12)(x + 5)
Setting each factor equal to zero:
x - 12 = 0 or x + 5 = 0
Solving for x, we have:
x = 12 or x = -5
Since a length cannot be negative, we can conclude that the longer leg of the triangle is 12 inches.
Now, we can substitute this value back into our equation to find the length of the shorter leg:
Shorter leg = longer leg - 7
Shorter leg = 12 - 7
Shorter leg = 5 inches
So, the side lengths of the right triangle are:
Longer leg = 12 inches
Shorter leg = 5 inches
Let's assume that the longer leg of the right triangle is represented by the variable x.
According to the given information, the shorter leg is 7 inches shorter than the longer leg. So, the length of the shorter leg can be represented as x - 7.
We can use the Pythagorean theorem to solve for the length of the sides. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Using this theorem, we can write the equation as follows:
(x - 7)^2 + x^2 = 13^2
Expanding the equation:
x^2 - 14x + 49 + x^2 = 169
Combining like terms:
2x^2 - 14x + 49 = 169
Rearranging the equation:
2x^2 - 14x - 120 = 0
Dividing through by 2 to simplify:
x^2 - 7x - 60 = 0
Factoring the quadratic equation:
(x - 12)(x + 5) = 0
Setting each factor equal to zero and solving for x:
x - 12 = 0 or x + 5 = 0
x = 12 or x = -5
Since the length of a side cannot be negative, we can discard x = -5.
Therefore, the longer leg of the right triangle is 12 inches.
Using this value, we can find the length of the shorter leg:
Shorter leg = x - 7 = 12 - 7 = 5 inches
So, the lengths of the sides of the triangle are as follows:
Longer leg = 12 inches
Shorter leg = 5 inches
Hypotenuse = 13 inches