Problem 77 - Elementary Algebra Within Reach

Question

In Exercises 77-80, evaluate the algebraic expression for the given values of the variable(s). Area of a Triangle \(\frac{1}{2} b h\) (a) \(b=3, h=5\) (b) \(b=2, h=10\)

Step-by-Step Solution

Answer

The areas of the triangles are 7.5 square units and 10 square units respectively.

Step 1: Identify the formula

The formula to find the area of a triangle is \(\frac{1}{2} \times b \times h\), where \(b\) is the base and \(h\) is the height of the triangle.

Step 2: Substitute the values into the formula (a)

For the first triangle, the base \(b\) is 3 and the height \(h\) is 5. Substitute these values into the formula: Area \(= \frac{1}{2} \times 3 \times 5\).

Step 3: Calculate the area of the first triangle (a)

Multiply the numbers together to find the area of the first triangle: Area = 7.5 square units.

Step 4: Substitute the values into the formula (b)

For the second triangle, the base \(b\) is 2 and the height \(h\) is 10. Substitute these values into the formula: Area \(= \frac{1}{2} \times 2 \times 10\).

Step 5: Calculate the area of the second triangle (b)

Multiply the numbers together to find the area of the second triangle: Area = 10 square units.

Key Concepts

Algebraic Expression

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They help us communicate mathematical ideas in a compact form. For instance, to compute the area of a triangle, we use the algebraic expression \(\frac{1}{2}bh\). Here, \(b\) and \(h\) are variables representing the base and height of the triangle. This expression means you should multiply the base by the height, then take half of the result to find the area. Understanding how to manipulate these expressions accurately is crucial for solving many real-world problems.

Triangle Formula

The triangle formula is used to find the area of a triangle, which is a measure of the space inside it. The specific formula is:

• Area \(= \frac{1}{2} \times b \times h\)

This formula gives the area by first multiplying the base \(b\) of the triangle by its height \(h\), and then dividing by 2. The division by 2 is necessary because a triangle is essentially half of a rectangle when you think about its base and height. Using the formula correctly requires knowing how to substitute the given values of \(b\) and \(h\) into it, then performing the calculation to find the area.

Base and Height

When determining the area of a triangle, identifying the base and height is key. The base \(b\) is one side of the triangle; it can be any one of the three sides. Once you pick a side as the base, the height \(h\) is the perpendicular distance from the base to the opposite vertex. The height must always form a right angle with the base.

• Base and height must be perpendicular to each other.
• They are used directly in the formula for the area.

Appropriately choosing the base and height ensures that your area calculation is correct. If any doubt arises in identifying the height, construct a perpendicular from the vertex opposite the base.

Multiplication in Algebra

Multiplication in algebra involves combining numbers and variables in specified ways to form products. In the context of the area of a triangle, you multiply three components:

• The fraction \(\frac{1}{2}\)
• The base \(b\)
• The height \(h\)

Firstly, it's helpful to multiply \(b\) and \(h\) to see the full area of a rectangle with the same base and height. Then, apply multiplication with the fraction \(\frac{1}{2}\) to obtain the area of the triangle. This step-by-step multiplication helps in minimizing mistakes and ensuring the area is calculated correctly. Mastery of multiplication, even when fractions are involved, is vital for solving algebraic expressions effectively.