the shortest side of a right triangle is 16 miles long. the difference between the length of the other two sides is 4 miles. find the missing sides. What is the longer leg? what is the hypotenuse?
by Pythagoras ... x^2 + 16^2 = (x + 4)^2
x^2 + 256 = x^2 + 8x + 16 ... 240 = 8x
Let's solve this step by step:
1. Let's assume the shorter leg of the right triangle is x miles.
2. According to the problem, the difference between the length of the other two sides is 4 miles. So, the longer leg would be x + 4 miles.
3. We know that one side of the triangle is 16 miles, which is the shortest side. Let's label it as the hypotenuse.
4. Now, we can use the Pythagorean theorem to find the missing sides. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, applying the theorem, we have:
16^2 = x^2 + (x + 4)^2.
Simplifying, we get:
256 = x^2 + (x^2 + 8x + 16).
Combining like terms:
256 = 2x^2 + 8x + 16.
Rearranging the equation:
2x^2 + 8x + 16 - 256 = 0.
2x^2 + 8x - 240 = 0.
5. We can divide both sides of the equation by 2 to simplify it, giving us:
x^2 + 4x - 120 = 0.
6. Now, we need to factorize the quadratic equation:
(x - 10)(x + 12) = 0.
This gives us two possible values for x: x = 10 or x = -12.
Since the length of a side cannot be negative, we discard the negative value. Therefore, x = 10 miles.
7. Now, we can find the longer leg:
Longer leg = x + 4 = 10 + 4 = 14 miles.
8. Finally, we can also find the hypotenuse:
Hypotenuse = 16 miles (given in the problem).
So the longer leg of the right triangle is 14 miles, and the hypotenuse is 16 miles.
To find the longer leg and the hypotenuse of the right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's denote the shorter leg as 'x'. Since we know that the difference between the lengths of the other two sides is 4 miles, we can express the longer leg as 'x + 4'.
According to the Pythagorean theorem, we can set up the following equation:
16^2 + x^2 = (x + 4)^2
Simplifying the equation:
256 + x^2 = x^2 + 8x + 16
Combine like terms:
256 = 8x + 16
Subtract 16 from both sides:
240 = 8x
Divide both sides by 8:
30 = x
Therefore, the shorter leg is 30 miles long.
To find the longer leg, we substitute the value of 'x' back into the expression 'x + 4':
30 + 4 = 34
So, the longer leg is 34 miles long.
Next, let's find the hypotenuse using the Pythagorean theorem. We know that the shortest side (the other leg) is 30 miles and the longer leg is 34 miles:
Hypotenuse^2 = 30^2 + 34^2
Hypotenuse^2 = 900 + 1156
Hypotenuse^2 = 2056
To find the hypotenuse, we take the square root of both sides:
Hypotenuse ≈ √2056
Using a calculator or estimating, we find that the hypotenuse is approximately 45.38 miles.
Therefore, the longer leg of the triangle is 34 miles, and the hypotenuse is approximately 45.38 miles.