The hypotenuse of a right triangle is 6 more than the shorter leg. The longer leg is 3 more than the shorter leg. What is the measure of the shorter leg of the right triangle?
If the shorter leg is x, then
x^2 + (x+3)^2 = (x+6)^2
You can do the math, or think of the familiar 3-4-5 right triangle, enlarged to 9-12-15
To find the measure of the shorter leg of the right triangle, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that the shorter leg of the right triangle is x. Using the given information, we can determine the lengths of the other sides as follows:
- The hypotenuse is 6 more than the shorter leg: Hypotenuse = x + 6
- The longer leg is 3 more than the shorter leg: Longer leg = x + 3
Now, we can apply the Pythagorean theorem:
(x + 6)^2 = x^2 + (x + 3)^2
Expanding and simplifying:
x^2 + 12x + 36 = x^2 + x^2 + 6x + 6x + 9
Combining like terms:
x^2 + 12x + 36 = 2x^2 + 12x + 9
Subtracting x^2 + 12x + 9 from both sides:
0 = x^2 - 27
Rearranging:
x^2 = 27
Taking the square root of both sides:
x = ±√27
Since we're dealing with a length, we take the positive square root:
x = √27
Simplifying the square root:
x = √(9 * 3) = 3√3
Therefore, the measure of the shorter leg of the right triangle is 3√3.
Let's assume the shorter leg of the right triangle is "x".
According to the problem, the longer leg is 3 more than the shorter leg, so it would be "x + 3".
Also, the hypotenuse is 6 more than the shorter leg, so it would be "x + 6".
By using the Pythagorean theorem for a right triangle (a^2 + b^2 = c^2), we can write the equation:
x^2 + (x + 3)^2 = (x + 6)^2
Expanding and simplifying the equation:
x^2 + (x^2 + 6x + 9) = x^2 + 12x + 36
x^2 + x^2 + 6x + 9 = x^2 + 12x + 36
Combining like terms:
2x^2 + 6x + 9 = x^2 + 12x + 36
Rearranging the equation:
x^2 - 6x - 27 = 0
Factoring the quadratic equation:
(x - 9)(x + 3) = 0
Setting each factor equal to zero:
x - 9 = 0 or x + 3 = 0
Solving for x:
x = 9 or x = -3
Since the length of a leg cannot be negative, the value of x is 9.
Therefore, the measure of the shorter leg of the right triangle is 9 units.