A right triangle has a height 8 cm more than twice the length of the base. If the area of the triangle is 96 cm2, what are the dimensions of the triangle.
b(2b+8)/2 = 96
solve for b, then get h, and the hypotenuse, if you want it.
Let's assume the length of the base of the right triangle is x cm.
According to the given information, the height of the triangle is 8 cm more than twice the length of the base. So, the height can be written as (2x + 8) cm.
The formula for the area of a triangle is (base * height) / 2. We have the area as 96 cm², so we can set up the equation:
(1/2) * x * (2x + 8) = 96
Now, let's solve this equation step-by-step.
Step 1: Distribute (1/2) into the expression (2x + 8):
x * (2x + 8) / 2 = 96
Step 2: Simplify the expression:
x * (2x + 8) = 96 * 2
Step 3: Expand the expression on the left side:
2x² + 8x = 192
Step 4: Subtract 192 from both sides of the equation:
2x² + 8x - 192 = 0
Step 5: Divide the entire equation by 2 to simplify it:
x² + 4x - 96 = 0
Step 6: Solve the quadratic equation either by factoring or using the quadratic formula. Let's use factoring.
The equation can be factored as:
(x + 12)(x - 8) = 0
Setting each factor equal to zero:
x + 12 = 0 or x - 8 = 0
Solving for x in both equations:
x = -12 or x = 8
Since we cannot have a negative value for the length of a side, we discard the solution x = -12.
Therefore, the length of the base is x = 8 cm.
Now, substitute the value of x into the expression for the height:
Height = 2x + 8
Height = 2 * 8 + 8
Height = 24 cm
So, the dimensions of the triangle are:
Base = 8 cm
Height = 24 cm
To solve this problem, we need to use the formula for the area of a triangle:
Area = (1/2) * base * height
Given that the area is 96 cm^2, we can write the equation as:
96 = (1/2) * base * height
We are also given that the height is 8 cm more than twice the base. Let's represent the base as "x".
So, the height is 2x + 8.
Substituting the values in the equation, we have:
96 = (1/2) * x * (2x + 8)
Multiplying both sides by 2 to remove the fraction, we get:
192 = x * (2x + 8)
Expanding the equation, we have:
192 = 2x^2 + 8x
Rearranging the equation, we get a quadratic equation:
2x^2 + 8x - 192 = 0
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 2, b = 8, and c = -192.
Plugging in the values, we get:
x = (-8 ± √(8^2 - 4 * 2 * -192)) / (2 * 2)
Simplifying, we get:
x = (-8 ± √(64 + 1536)) / 4
x = (-8 ± √1600) / 4
x = (-8 ± 40) / 4
Now, we have two possible values for x:
1) x = (-8 + 40) / 4 = 32 / 4 = 8
2) x = (-8 - 40) / 4 = -48 / 4 = -12
Since the length cannot be negative, we discard the second solution and take x = 8.
Therefore, the dimensions of the right triangle are:
Base = 8 cm
Height = 2*8 + 8 = 24 cm
So, the dimensions of the triangle are 8 cm for the base and 24 cm for the height.