A triangle has a height that is 4 inches less than its base. If the triangle has an area of 86 square inches, set up and resolve an equation to find its base and height.
(b*(b-4))/2 = 86
Let's assume the base of the triangle is represented by 'b' inches.
According to the given information, the height of the triangle is 4 inches less than the base. So, the height can be represented as 'b - 4' inches.
The formula for the area of a triangle is given by:
Area = (base * height) / 2
Substituting the given values:
86 = (b * (b - 4)) / 2
To solve for 'b', we can multiply both sides of the equation by 2:
2 * 86 = b * (b - 4)
Simplifying this equation:
172 = b^2 - 4b
Rearranging the equation:
b^2 - 4b - 172 = 0
Now we have a quadratic equation. We can solve this either by factoring or by using the quadratic formula. Let's use the quadratic formula to find 'b':
b = (-(-4) ± √((-4)^2 - 4 * 1 * -172)) / (2 * 1)
Simplifying further:
b = (4 ± √(16 + 688)) / 2
b = (4 ± √(704)) / 2
b = (4 ± 26.52) / 2
Now we have two possible values for 'b':
1. b = (4 + 26.52) / 2 = 15.76 inches
2. b = (4 - 26.52) / 2 = -11.76 inches (since it doesn't make sense in this context, we can ignore this negative value)
Therefore, the base of the triangle is approximately 15.76 inches.
To find the height, we can substitute the value of 'b' into the expression 'b - 4':
height = 15.76 - 4 = 11.76 inches
Therefore, the height of the triangle is approximately 11.76 inches.
To solve this problem, we can first define the variables we need. Let's use "b" to represent the base of the triangle and "h" to represent the height of the triangle.
From the problem statement, we know that the height of the triangle is 4 inches less than its base. So we can express the height as:
h = b - 4
We also know that the area of the triangle is 86 square inches, and the formula for the area of a triangle is given by:
Area = (1/2) * base * height
Substituting the given values, we have:
86 = (1/2) * b * (b - 4)
To solve for the base of the triangle, we can simplify and rearrange the equation. Let's start by multiplying both sides of the equation by 2 to get rid of the fraction:
2 * 86 = b * (b - 4)
172 = b^2 - 4b
Now, we can move all the terms to one side of the equation to obtain a quadratic equation:
b^2 - 4b - 172 = 0
We can now solve this quadratic equation to find the value(s) of b, which represents the base of the triangle. We can factor the equation or use the quadratic formula to solve it.
Using the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -4, and c = -172. Substituting these values into the formula, we have:
b = (-(-4) ± √((-4)^2 - 4(1)(-172))) / (2(1))
Simplifying further, we get:
b = (4 ± √(16 + 688)) / 2
b = (4 ± √(704)) / 2
b = (4 ± √(16 * 44)) / 2
b = (4 ± 4√11) / 2
Now, we can simplify this expression by canceling out the common factors:
b = 2 ± 2√11
Therefore, the base of the triangle can have two possible values: 2 + 2√11 and 2 - 2√11.
To find the height of the triangle, we can substitute the values of b into the equation for the height:
h = b - 4
h1 = (2 + 2√11) - 4
h1 = -2 + 2√11
h2 = (2 - 2√11) - 4
h2 = -2 - 2√11
So, the height of the triangle can have two possible values: -2 + 2√11 and -2 - 2√11.
However, since a height cannot be negative in this context, we discard the negative values. Therefore, the base of the triangle is 2 + 2√11 inches, and the height is -2 + 2√11 inches.