Problem 32 - Algebra A Combined Function

Question

The height of a triangle is 2 millimeters less than the base. If the area is 60 square millimeters, find the base.

Step-by-Step Solution

Answer

The base of the triangle is 12 millimeters.

Step 1: Understand the Problem

We are given a triangle where the height is 2 millimeters less than the base, and the area is 60 square millimeters. We need to find the value of the base.

Step 2: Establish Relationships

Let the base of the triangle be \( b \) millimeters. Then, the height of the triangle is \( b - 2 \) millimeters, since it is 2 millimeters less than the base.

Step 3: Use the Area Formula for a Triangle

The area \( A \) of a triangle is given by \( A = \frac{1}{2} \times \text{base} \times \text{height} \). For our specific triangle, this becomes:\[ 60 = \frac{1}{2} \times b \times (b - 2) \]

Step 4: Solve the Equation for Base

Simplify and solve the equation:\[\begin{align*}60 &= \frac{1}{2} \times b \times (b - 2) \120 &= b \times (b - 2) \120 &= b^2 - 2b \b^2 - 2b - 120 &= 0\end{align*}\]Solve this quadratic equation using the quadratic formula, \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1 \), \( B = -2 \), and \( C = -120 \).

Step 5: Apply Quadratic Formula

Calculate the discriminant:\[ B^2 - 4AC = (-2)^2 - 4 \times 1 \times (-120) = 4 + 480 = 484 \]Now solve for \( b \):\[ b = \frac{-(-2) \pm \sqrt{484}}{2 \times 1} \]\[ b = \frac{2 \pm 22}{2} \]Calculate the two possibilities for \( b \):1. \( b = \frac{2 + 22}{2} = 12 \)2. \( b = \frac{2 - 22}{2} = -10 \)Since the base cannot be negative, \( b = 12 \) millimeters.

Key Concepts

Area of Triangle Formula

The area of a triangle is a fundamental concept in geometry, and it helps in solving various geometric problems.
To find the area of a triangle, you need to use the formula:

• \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

The "base" is any one of the triangle's sides, and the "height" is the perpendicular distance from the base to the opposite vertex.
This formula works because a triangle is essentially half of a parallelogram with the same base and height. Knowing how to manipulate this formula allows for finding an unknown side or solving for other properties of the triangle.
In our specific problem, knowing the area and the relationship between base and height helped set up the equation necessary to find the unknown base.

Quadratic Formula

A quadratic equation is any equation that can be put into the form of \( ax^2 + bx + c = 0 \).
These types of equations can be solved using the quadratic formula:

• \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

The quadratic formula is a critical tool because it provides a solution for all types of quadratic equations, regardless of whether they can be factored easily.
To use it, you identify the coefficients \( A \), \( B \), and \( C \) from the quadratic equation and plug them into the formula.
In our case for the base of the triangle problem, we rearranged the equation from the area formula into a standard quadratic equation form, enabling us to solve for the base using this formula.

Discriminant Calculation

The discriminant is part of the quadratic formula under the square root symbol, given by \( B^2 - 4AC \).
It determines the nature of the roots of a quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root.
• If it's negative, the equation has no real roots, just complex ones.

In solving the triangle problem, the discriminant came out to be 484, which is positive.
This indicated that the quadratic equation had two real roots.By calculating the roots, we found that the base of the triangle was 12 millimeters, discarding the negative root as an unrealistic solution for a length.