The base of a triangle is three inches more than its height. If each is increased by 3 inches the
area of the triangle (1/2 bh)
is 14 square inches. Find the original base (𝑏) and the original height
(ℎ) in inches. (Lesson 15)
To find the original base and height of the triangle, we can use the information given in the problem. Let's break it down step by step.
1. Let's assume the height of the triangle is represented by "h" inches. According to the problem, the base of the triangle is three inches more than the height. So the base can be represented as "h + 3" inches.
2. The problem also states that if each, the base and the height, is increased by 3 inches, the area of the triangle becomes 14 square inches. Let's calculate the new base and height.
New base = (h + 3) + 3 = h + 6 inches
New height = h + 3 inches
3. The formula for the area of a triangle is given as (1/2) * base * height. We are told that the new area is 14 square inches, so we can set up an equation using the new base and height:
(1/2) * (h + 6) * (h + 3) = 14
4. Now, we can solve this equation to find the value of "h." We will multiply both sides of the equation by 2 to eliminate the fraction:
(h + 6) * (h + 3) = 28
Expanding the equation:
h^2 + 9h + 18 = 28
Rearranging the terms:
h^2 + 9h - 10 = 0
5. To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring is a bit challenging, so let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 1, b = 9, and c = -10. Substituting these values into the formula:
h = (-9 ± √(9^2 - 4 * 1 * -10))/(2 * 1)
Simplifying:
h = (-9 ± √(81 + 40))/2
h = (-9 ± √121)/2
h = (-9 ± 11)/2
This gives us two possible values for "h":
h1 = (-9 + 11)/2 = 1
h2 = (-9 - 11)/2 = -10/2 = -5
Since a height cannot be negative, we'll discard the negative value of "h."
6. Now that we have the height "h" as 1 inch, we can substitute it back into the base equation to find the original base:
Base (b) = h + 3 = 1 + 3 = 4 inches
Thus, the original base of the triangle is 4 inches and the original height is 1 inch.