The shortest leg of a triangle is 2 inches shorter than the other leg. The hypotenuse of this triangle is 10 inches. What are the lengths of the two legs of this triangle?
x^2 + (x + 2)^2 = 10^2
2 x^2 + 4 x + 4 = 100 ... x^2 + 2x + 2 = 50
solve for x (the shortest leg)
idk
Let's assume the length of the shorter leg is x inches.
According to the given information, the length of the longer leg is x + 2 inches.
Using the Pythagorean theorem, we know that the sum of the squares of the two legs is equal to the square of the hypotenuse:
(shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2
x^2 + (x + 2)^2 = 10^2
Expanding the equation, we get:
x^2 + (x^2 + 4x + 4) = 100
Combining like terms, we have:
2x^2 + 4x + 4 = 100
Moving all terms to one side of the equation, we have:
2x^2 + 4x - 96 = 0
Dividing the equation by 2, we get:
x^2 + 2x - 48 = 0
Factoring the equation, we have:
(x - 6)(x + 8) = 0
Setting each factor equal to zero and solving for x, we find two possible values for x: x = 6 and x = -8.
Since we are dealing with the length of a side of a triangle, a negative value is not meaningful. Therefore, the length of the shorter leg is x = 6 inches.
The length of the longer leg is x + 2 = 6 + 2 = 8 inches.
Thus, the lengths of the two legs of the triangle are 6 inches and 8 inches.
To find the lengths of the two legs of the triangle, we can set up a system of equations based on the given information.
Let x represent the length of the longer leg.
Then the length of the shorter leg would be (x - 2), as it is 2 inches shorter than the longer leg.
According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.
So, we have:
x^2 + (x - 2)^2 = 10^2
Expanding and simplifying:
x^2 + (x^2 - 4x + 4) = 100
2x^2 - 4x + 4 = 100
2x^2 - 4x - 96 = 0
Now, we can solve this quadratic equation to find the values of x.
Using factoring, we can factor out a common factor of 2:
2(x^2 - 2x - 48) = 0
Now, we have:
x^2 - 2x - 48 = 0
Factoring the quadratic equation:
(x - 8)(x + 6) = 0
Setting each factor equal to zero and solving for x:
x - 8 = 0 => x = 8
x + 6 = 0 => x = -6
Since the length of a side cannot be negative, we ignore the x = -6 solution.
Therefore, the length of the longer leg (x) is 8 inches, and the length of the shorter leg (x - 2) is (8 - 2) = 6 inches.
So, the lengths of the two legs of this triangle are 6 inches and 8 inches.