one leg of a right triangle is 9 inches shorter than the other leg. the hypotenuse is 45 inches long how long is the shorter of the legs?
x^2 + (x-9)^2 = 45^2
just guessing at a 3-4-5 triangle, try
27-36-45
yep. 27 is 9 less than 36.
To find the length of the shorter leg in a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's call the length of the shorter leg "x" inches. According to the problem, the other leg is 9 inches longer, so its length would be "x + 9" inches.
Now, we can set up the equation based on the Pythagorean theorem:
x^2 + (x + 9)^2 = 45^2
Expanding and simplifying the equation:
x^2 + (x^2 + 18x + 81) = 2025
Combining like terms:
2x^2 + 18x + 81 - 2025 = 0
2x^2 + 18x - 1944 = 0
Now, we can solve this quadratic equation using various methods, such as factoring, completing the square, or using the quadratic formula.
One way to solve this equation is by factoring, if possible. Factoring the equation, we get:
(2x - 36)(x + 54) = 0
Setting each factor equal to zero:
2x - 36 = 0 or x + 54 = 0
Solving each equation separately:
2x = 36 or x = -54
Dividing by 2:
x = 18 or x = -54
Since length cannot be negative, we discard x = -54.
Therefore, the length of the shorter leg, x, is 18 inches.