Final Answer:
The base of the triangle is 20 inches and the altitude is 16 inches.
Step-by-step explanation:
To solve this problem, let us denote the base of the triangle as "b" and let the altitude be "b - 4" since we know that the altitude is four inches less than the base.
The area of a triangle can be calculated using the formula:
Area = 1/2 × base × altitude
Given that the area of the triangle is 160 square inches, we can set up the equation:
160 = 1/2 × b × (b - 4)
Expanding the right side and multiplying both sides by 2 to eliminate the fraction, we get:
320 = b × (b - 4)
320 = b² - 4b
Now, we need to rearrange the equation into a standard quadratic form:
b² - 4b - 320 = 0
To solve this quadratic equation, we can factor it, use the quadratic formula, or complete the square. Factoring seems to be the simplest approach here, so let's attempt that. We need to find two numbers that multiply to -320 and add up to -4.
By inspection or trial and error, we find that 16 and -20 satisfy these requirements because:
16 × (-20) = -320
16 + (-20) = -4
Thus, we can factor the quadratic equation as follows:
(b - 20)(b + 16) = 0
Setting each factor equal to zero gives us two possible solutions for b:
b - 20 = 0 or b + 16 = 0
b = 20 or b = -16
Since the base of a triangle cannot be negative, we reject the solution b = -16. Therefore, the base of the triangle is 20 inches.
Now that we have the value for the base, we can find the altitude by subtracting 4 from the base:
Altitude = b - 4 = 20 - 4 = 16 inches
So, the base of the triangle is 20 inches and the altitude is 16 inches.