The hypotenuse of a right triangle is 22 m long. The length of one leg is 10 m less than the other. Find the lengths of the legs.
I have so far
let one leg be x
the other leg will be x-10
x^2 +(x-10)^2=22^2
x^2+x^2-20x+100=484
carry on ...
x^2+x^2-20x+100=484
2x^2 - 20x - 384 = 0
x^2 - 10x - 192 = 0
I used the formula to solve for x
x = 19.7309
one side is 19.7309, the other 9.7309
x^2+x^2-20x+100=484
--> 2x^2-20x-384=0
--> x^2-10x-192=0
Now apply quadratic formula.
To solve the equation, you can combine like terms on the left side:
2x^2 - 20x + 100 = 484
Next, subtract 484 from both sides:
2x^2 - 20x - 384 = 0
Now, we have a quadratic equation in standard form. To solve it, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -20, and c = -384. Plugging these values into the quadratic formula, we can find the values of x.
x = (-(-20) ± √((-20)^2 - 4*2*(-384))) / (2*2)
Simplifying further, we have:
x = (20 ± √(400 + 3072)) / 4
x = (20 ± √(3472)) / 4
Now, you can calculate the two possible values of x by performing the calculations inside the square root:
x = (20 ± √(3472)) / 4
x ≈ (20 ± 58.92) / 4
Now, you have two possible values for x:
x1 ≈ (20 + 58.92) / 4 = 78.92 / 4 ≈ 19.73
x2 ≈ (20 - 58.92) / 4 = -38.92 / 4 ≈ -9.73
Since the length of a side cannot be negative, the only valid solution for x is x ≈ 19.73.
Now, you can find the lengths of the legs:
The length of the first leg = x ≈ 19.73 m
The length of the second leg = x - 10 ≈ 19.73 - 10 ≈ 9.73 m
Therefore, the lengths of the legs are approximately 19.73 m and 9.73 m.