The height of a triangle is 8 m more than twice the length of the base. The area of the triangle is 32 m^2. Whats the height of the triangle
The height of a triangle is 8 m more than twice the length of the base. The area of the triangle is 21 m2. Find the height of the triangle.
To find the height of a triangle, we can use the formula for the area of a triangle. Area = (1/2) * base * height.
Given that the area of the triangle is 32 m^2, we can substitute this value into the formula:
32 = (1/2) * base * height
We are also given that the height of the triangle is 8 m more than twice the length of the base. So, we can express the height as:
height = 2 * base + 8
Now we can substitute this expression for height into the area formula:
32 = (1/2) * base * (2 * base + 8)
Simplifying this equation will allow us to find the value of base, which we can then use to calculate the height.
Let's proceed with the simplification of the equation:
32 = (1/2) * (2 * base^2 + 8 * base)
Multiplying both sides of the equation by 2 to remove the fraction:
64 = 2 * base^2 + 8 * base
Rearranging the equation and setting it equal to zero:
2 * base^2 + 8 * base - 64 = 0
Now we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 8, and c = -64.
To solve this equation, we can factor it or use the quadratic formula. Factoring this equation might not be straightforward, so let's use the quadratic formula instead:
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c:
base = (-8 ± √(8^2 - 4 * 2 * -64)) / (2 * 2)
Simplifying inside the square root:
base = (-8 ± √(64 + 512)) / 4
base = (-8 ± √576) / 4
base = (-8 ± 24) / 4
This gives us two possible values for the base. We'll calculate both:
base₁ = (-8 + 24) / 4 = 16 / 4 = 4
base₂ = (-8 - 24) / 4 = -32 / 4 = -8
Since the length of a base cannot be negative, we discard the negative value for base and conclude that the base of the triangle is 4 m.
Now we can substitute this value back into the expression for the height:
height = 2 * base + 8 = 2 * 4 + 8 = 16
Therefore, the height of the triangle is 16 meters.
Area = 1/2 Height * Base
If Height = x, then base = 2x+8
32 = 1/2x(2x+8)
Solve for x.