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.
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.
b(2b+8)/2 = 96
solve for b, then get h, and the hypotenuse, if you want it.