A right triangle has hypotenuse 5 inches long. Its area is 4 square inches. How long are the other sides?
1/2 bh=4
bh=8
b^2+h^2=25
h^2/64 +h^2=25
65h^2=25*64
h= 5*8/sqrt65
I don't quite follow the last three lines. Could you please explain?
To solve this problem, we can use the Pythagorean theorem and the formula for the area of a right triangle.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, 5 inches) is equal to the sum of the squares of the lengths of the other two sides.
So, let's assume the lengths of the other two sides are represented by variables a and b. We can set up the equation as follows:
a^2 + b^2 = 5^2
We are also given the area of the triangle as 4 square inches. The formula for the area of a right triangle is:
Area = (1/2) * base * height
Since the triangle is a right triangle, one of the sides can be considered the base, and the other side can be considered the height. So, we have:
Area = (1/2) * a * b
Substituting the given area (4 square inches) into the equation, we get:
4 = (1/2) * a * b
Now, we have a system of two equations:
a^2 + b^2 = 5^2 ----(1)
4 = (1/2) * a * b ----(2)
To solve this system of equations, we can substitute the value of "a * b" from equation (2) into equation (1).
From equation (2), we can rewrite it as:
2 = (a * b) / 4
Multiplying both sides by 4, we get:
8 = a * b
Now, we can substitute this value into equation (1):
a^2 + b^2 = 5^2
(a * b) + (a * b) = 5^2 [Substituting 8 for a * b]
2 * (a * b) = 5^2
2 * 8 = 5^2
16 = 25
This is not true, which means there is no solution to the system of equations.
Hence, there are no lengths for the other two sides of the right triangle that satisfy the given conditions.