the area of a right triangle is 12 in ^2. the ratio of the lengths of its legs is 2/3. find the hypotenuse.
Let the shortest leg length be a.
Area = (a)*(3a/2)*(1/2) = (3/4)a^2 = 12
a^2 = 16
a = 4 and the other leg (which we'll call b) must be 6.
Use the Pythagorean theorem for the hypotenuse, c.
c^2 = a^2 + b^2
c = sqrt(52) = 7.2111 inch
To find the hypotenuse of a right triangle, 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 other two sides.
Let's assume the lengths of the legs of the right triangle are 2x and 3x (since the ratio of their lengths is given as 2/3).
According to the Pythagorean theorem, we have:
(2x)^2 + (3x)^2 = hypotenuse^2
Expanding and simplifying:
4x^2 + 9x^2 = hypotenuse^2
13x^2 = hypotenuse^2
Now, we are given that the area of the right triangle is 12 square inches. The area of a triangle can be calculated using the formula:
Area = 1/2 * base * height
Since the triangle is a right triangle, the base and height are the lengths of the legs. So we have:
1/2 * 2x * 3x = 12 square inches
6x^2 = 12
Dividing by 6, we get:
x^2 = 2
Taking the square root of both sides, we have:
x = √2
Now we can substitute this value of x back into the equation to find the hypotenuse:
hypotenuse^2 = 13x^2
hypotenuse^2 = 13(√2)^2
hypotenuse^2 = 13 * 2
hypotenuse^2 = 26
Taking the square root of both sides, we have:
hypotenuse = √26
So the length of the hypotenuse of the right triangle is √26 inches.
To find the hypotenuse of a right triangle given the area and the ratio of the lengths of its legs, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Let's label the lengths of the legs as b and h, with b being the base and h being the height. We're given that the ratio of the lengths of the legs is 2/3, so we can represent them as follows:
b = (2/3)h
Since the area is given as 12 square inches, we can substitute these values into the area formula:
12 = (1/2) * b * h
Substituting (2/3)h for b:
12 = (1/2) * (2/3)h * h
Simplifying the equation:
12 = (1/3) * h^2
To isolate h^2, multiply both sides of the equation by 3:
36 = h^2
Taking the square root of both sides of the equation to solve for h:
h = √36
h = 6
Now that we have the height (h) as 6 inches, we can find the base (b) using the ratio:
b = (2/3) * 6
b = 4
Finally, we can use the Pythagorean theorem to find the hypotenuse (c):
c^2 = b^2 + h^2
c^2 = 4^2 + 6^2
c^2 = 16 + 36
c^2 = 52
Taking the square root of both sides of the equation to solve for c:
c = √52
c ≈ 7.21 inches
So, the hypotenuse of the right triangle is approximately 7.21 inches.