How do I solve: One leg of a right triangle is 4 m longer than the other. If the length of the hypotenuse is 18 m, how long are the other two legs?
x^2 + (x+4)^2 = 18^2 = 324
2 x^2 + 8 x + 16 = 324
2 x^2 + 8 x - 308 = 0
x^2 + 4 x -154 = 0
x = [ -4 +/- sqrt (16 + 616)]/2
= [-4 +/- 2 sqrt 158 ] /2
= -2 +/- sqrt 158 use + root
10.57
14.57
To solve this problem, we can use the Pythagorean theorem, 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.
Let's assume that the length of the shorter leg is x. According to the given information, the length of the other leg is x + 4.
Using the Pythagorean theorem, we can set up the following equation:
x^2 + (x + 4)^2 = 18^2
Expanding the equation, we get:
x^2 + (x^2 + 8x + 16) = 324
Combining like terms, we have:
2x^2 + 8x + 16 = 324
Rearranging the equation, we get:
2x^2 + 8x - 308 = 0
To solve for x, we can factor the quadratic equation:
2(x^2 + 4x - 154) = 0
(x + 14)(x - 10) = 0
Setting each factor equal to zero, we get:
x + 14 = 0 or x - 10 = 0
x = -14 or x = 10
Since the length of a side cannot be negative, we can discard x = -14 as an extraneous solution.
Thus, the length of the shorter leg is 10 m.
To find the length of the longer leg, we add 4 to the length of the shorter leg:
x + 4 = 10 + 4 = 14 m
Therefore, the lengths of the two legs are 10 m and 14 m, respectively.
To solve this problem, we can start by assigning variables to the lengths of the legs. Let's say the length of one leg is 'x' meters. Since the problem states that one leg is 4 meters longer than the other leg, we can express the other leg's length as 'x + 4' meters.
Using the Pythagorean theorem, we can relate the lengths of the legs and the hypotenuse of a right triangle. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, in this case, we can write the equation:
(x)^2 + (x + 4)^2 = (18)^2
Now, let's solve this equation step by step:
Expanding the equation:
x^2 + (x + 4)(x + 4) = 324
Simplifying:
x^2 + (x^2 + 8x + 16) = 324
Combining like terms:
2x^2 + 8x + 16 = 324
Rearranging the equation:
2x^2 + 8x + 16 - 324 = 0
Simplifying:
2x^2 + 8x - 308 = 0
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's solve it by factoring:
Dividing the equation by 2 to simplify it:
x^2 + 4x - 154 = 0
Factoring the simplified equation:
(x + 14)(x - 11) = 0
Setting each factor to zero:
x + 14 = 0 or x - 11 = 0
Solving each equation separately:
x = -14 or x = 11
Since length cannot be negative, we can ignore the negative value. Therefore, x = 11.
So, one leg of the right triangle is 11 meters, and the other leg is 11 + 4 = 15 meters.
To summarize:
Length of one leg = 11 meters
Length of the other leg = 15 meters