the hypotenuse of a right triangle is 25 inches and the height is 5 inches more than its base. what is the height of the triangle?
25^2=H^2+(h-5)^2
solve for height h
You take the square root for both sides to cancel out the square. Then you solve for H. After finding H, you add 5 to find the height, which is 20
Let's solve this step-by-step.
Step 1: Let's denote the base of the triangle as "x" inches.
Step 2: Since the height is 5 inches more than the base, we can write it as "x + 5" inches.
Step 3: According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, it can be written as:
x^2 + (x + 5)^2 = 25^2
Step 4: Simplifying the equation:
x^2 + (x^2 + 10x + 25) = 625
Step 5: Combine like terms:
2x^2 + 10x + 25 = 625
Step 6: Move 625 to the other side of the equation:
2x^2 + 10x + 25 - 625 = 0
2x^2 + 10x - 600 = 0
Step 7: Factor the quadratic equation:
2(x^2 + 5x - 300) = 0
Step 8: Solve the quadratic equation:
x^2 + 5x - 300 = 0
(x + 20)(x - 15) = 0
Step 9: Set each factor to zero and solve for x:
x + 20 = 0 or x - 15 = 0
x = -20 or x = 15
Since the base cannot be negative, we discard -20 as a solution.
Step 10: The base of the triangle is 15 inches.
Step 11: The height of the triangle is x + 5 = 15 + 5 = 20 inches.
Therefore, the height of the triangle is 20 inches.
To find the height of the triangle, let's start by assigning variables to the base and height.
Let's say the base of the triangle is "x" inches.
According to the problem, the height is 5 inches more than the base. So, the height can be represented as "x + 5" inches.
Since we have the hypotenuse of the triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle.
In this case, the hypotenuse is 25 inches, and the base and height are "x" inches and "x + 5" inches, respectively.
Using the Pythagorean theorem, we can write the equation as follows:
(x^2) + (x + 5)^2 = 25^2
Now, let's solve this equation to find the value of "x" (which represents the base of the triangle).
Expanding the equation:
x^2 + (x^2 + 10x + 25) = 625
Combining like terms:
2x^2 + 10x + 25 = 625
Rearranging the equation to have zero on one side:
2x^2 + 10x + 25 - 625 = 0
2x^2 + 10x - 600 = 0
Now, we can solve this quadratic equation for "x" using factoring, completing the square, or the quadratic formula. In this case, let's solve it using factoring:
Finding two numbers whose product is -600 and whose sum is 10:
(x + 30) (x - 20) = 0
Setting each factor to zero:
x + 30 = 0 or x - 20 = 0
Solving for x:
x = -30 or x = 20
Since the length cannot be negative, we discard the negative value:
x = 20
Therefore, the base of the triangle is 20 inches.
To find the height, we can substitute the value of "x" into the equation we established earlier:
Height = x + 5
Height = 20 + 5
Height = 25 inches
Therefore, the height of the triangle is 25 inches.