The shorter leg of a right triangle is
6 inches shorter than the longer leg. The hypotenuse is 6 inches longer than the longer leg. Find the side lengths of the triangle.
(x-6)^2 + x^2 = (x+6)^2
or, consider that a 3-4-5 triangle has sides which differ by 1. You have a triangle whose sides differ by 6.
To find the side lengths of the triangle, let's start by assigning variables to the unknown lengths.
Let's say that the longer leg of the triangle is represented by the variable "x" inches.
According to the problem, the shorter leg is 6 inches shorter than the longer leg. So, the length of the shorter leg can be represented as "x - 6" inches.
We are also given that the hypotenuse is 6 inches longer than the longer leg. Therefore, the length of the hypotenuse can be represented as "x + 6" inches.
Since the triangle is a right triangle, we can use the Pythagorean theorem to solve for the side lengths.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
Using this information, we can set up the following equation:
(x - 6)^2 + x^2 = (x + 6)^2
Expanding and simplifying the equation:
x^2 - 12x + 36 + x^2 = x^2 + 12x + 36
Rearranging the equation:
2x^2 - 12x + 36 = x^2 + 12x + 36
Combining like terms:
x^2 - 24x = 0
Factoring out an "x":
x(x - 24) = 0
Setting each factor equal to zero and solving for "x":
x = 0 or x - 24 = 0
Since we are looking for the length of one leg of a triangle, we can disregard the solution x = 0.
Thus, we find that x = 24.
So, the longer leg of the triangle is 24 inches.
Using this value, we can calculate the lengths of the other sides.
The shorter leg = x - 6 = 24 - 6 = 18 inches
The hypotenuse = x + 6 = 24 + 6 = 30 inches
Therefore, the side lengths of the triangle are:
Longer leg = 24 inches
Shorter leg = 18 inches
Hypotenuse = 30 inches
Let's assume that the longer leg of the right triangle is represented by the variable "x" inches.
According to the problem, the shorter leg is 6 inches shorter than the longer leg. So, the length of the shorter leg can be represented as "x - 6" inches.
Also, the hypotenuse is 6 inches longer than the longer leg. Therefore, the length of the hypotenuse can be represented as "x + 6" inches.
Now, we can use the Pythagorean theorem to solve for the side lengths of the triangle, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Using this theorem, we can set up the following equation:
(x + 6)^2 = x^2 + (x - 6)^2
Expanding both sides of the equation:
x^2 + 12x + 36 = x^2 + x^2 - 12x + 36
Combining like terms:
x^2 + 12x + 36 = 2x^2 - 12x + 36
Subtracting x^2 and 36 from both sides:
12x = x^2
Dividing both sides by x:
12 = x
Therefore, the longer leg of the triangle is 12 inches.
Substituting the value of x back into the other expressions:
Shorter leg: x - 6 = 12 - 6 = 6 inches
Hypotenuse: x + 6 = 12 + 6 = 18 inches
Therefore, the side lengths of the triangle are:
Longer leg: 12 inches
Shorter leg: 6 inches
Hypotenuse: 18 inches