The length of the base and the height of a triangle are numerically equal. Their sum is 16 less than the number of units in the area of the triangle. Find the area of the triangle.
let the height be x
then the base is x
area of triangle = (1/2)x^2
sum = 2x
2x = (1/2)x^2 - 16
4x = x^2 - 32
x^2 - 4x - 32 = 0
(x-8)(x+4) = 0
x = 8 or x = -4, the latter makes no sense,
so if x=8, area = (1/2)64 = 32
check
is their sum 16 less than the area?
or
is 16 less than 32 by 16 ? , YES
To solve this problem, let's break it down step by step.
Let's assume the length of the base of the triangle is 'x' units.
Since the length of the base and the height of the triangle are numerically equal, the height of the triangle is also 'x' units.
The formula to calculate the area of a triangle is given by: Area = (base * height) / 2
Substituting 'x' for both the base and height, we get:
Area = (x * x) / 2
Area = x^2 / 2
The problem states that the sum of the base and the height is 16 less than the number of units in the area of the triangle. In other words, (base + height) = Area - 16.
Substituting the values, we have:
x + x = x^2 / 2 - 16
Multiplying both sides of the equation by 2 to eliminate the fraction, we get:
2x + 2x = x^2 - 32
Simplifying the equation:
4x = x^2 - 32
Rearranging to bring all the terms to one side:
x^2 - 4x - 32 = 0
Now, we have a quadratic equation. We can solve it by factoring or using the quadratic formula.
Factoring the equation, we have:
(x - 8)(x + 4) = 0
Setting each factor equal to zero, we get two possible solutions:
x - 8 = 0 or x + 4 = 0
Solving for 'x', we have:
x = 8 or x = -4
Since the dimensions of the triangle cannot be negative, we discard the solution x = -4.
Therefore, the length of the base of the triangle is 8 units.
To find the area, we substitute the value of 'x' back into the area formula:
Area = 8^2 / 2
Area = 64 / 2
Area = 32 square units
Hence, the area of the triangle is 32 square units.