The height of a triangle is 1 in. more than twice the base. Find the base and the height if the area of the triangle is 52-1/2 sq in. (Please provide steps/explanation to solve this problem.)
lets the height of the triangle h, and the base is b
first sentence means h = 1 + 2b
Area of the triangle formula is
A = 1/2(b*h)
since h = 1 + 2b
then A = 1/2 [ b * (1+2b)]
then you just need to replace the A with the given number and solve for b. After you find b, substitute b into h = 1+ 2b to find h
good luck
Thanks for the help. :)
To solve this problem, let's carefully analyze the given information.
We know that the area of a triangle is given by the formula:
Area = (1/2) * base * height
We are given that the area of the triangle is 52 1/2 sq in. So we can set up the equation as follows:
52 1/2 = (1/2) * base * height
Now, let's proceed with solving the equation step by step:
1. We need to express 52 1/2 as a fraction. It can be written as 105/2.
105/2 = (1/2) * base * height
2. We are also given that the height of the triangle is 1 in. more than twice the base. We can express this relationship as follows:
height = 2(base) + 1
Substitute this into the equation:
105/2 = (1/2) * base * (2(base) + 1)
3. Simplify the equation:
105 = base * (2(base) + 1) [Multiply both sides by 2 to eliminate the fractions]
4. Expand the equation:
105 = 2(base)^2 + base
5. Rearrange the equation into a quadratic form:
2(base)^2 + base - 105 = 0
6. Now we have a quadratic equation:
2(base)^2 + base - 105 = 0
We can solve this by factoring or using the quadratic formula.
Let's use factoring here:
(2(base) - 7)(base + 15) = 0
Therefore, either 2(base) - 7 = 0 or base + 15 = 0
Solve each equation separately:
2(base) - 7 = 0 -> 2(base) = 7 -> base = 7/2 -> base = 3.5
base + 15 = 0 -> base = -15
Since we cannot have a negative value for the base, we discard the second solution.
7. Now that we have the base, we can substitute it back into the expression for the height:
height = 2(base) + 1
height = 2(3.5) + 1
height = 7 + 1
height = 8
Thus, the base of the triangle is 3.5 inches and the height is 8 inches.