Problem 88 - Algebra 1: Concepts and Skills

Question

Find the area of a triangle with the given base and height. $$b=12, h=9$$

Step-by-Step Solution

Answer

The area of the triangle is 54 square units.

Step 1: Identify Given Values

Identify the given base and height values. Here, the base \(b=12\) and the height \(h=9\).

Step 2: Apply the Area Formula for a Triangle

Use the formula for the area of a triangle, which is \(\frac{1}{2} * base*width\). Substitute the given base and height values into the formula to calculate the area.

Step 3: Calculate the Area

Substitute the given base and height values \(b=12\) and \(h=9\) into the formula. \[Area = \frac{1}{2} * 12 * 9 = 54\] square units.

Key Concepts

Base and Height of a Triangle

When calculating the area of a triangle, two important dimensions need to be considered: the base and the height. The base (\(b\)) is any one of the triangle's three sides, while the height (\(h\)) is the perpendicular distance from the base to the opposite vertex. It's crucial to ensure accuracy when measuring or identifying these dimensions, as they can significantly influence area calculations. Keep in mind that the height is always taken at a right angle to the base, intersecting it at a 90-degree angle.

• The base can be any side of the triangle, though it is conventionally horizontal in illustrations.
• The height must connect with the base at a right angle.

By correctly identifying these two measurements, one can confidently proceed in calculating the triangle's area.

Triangle Area Formula

The formula for finding the area of a triangle is a fundamental concept in geometry. The area (\(A\)) of a triangle can be calculated using the formula:\[A = \frac{1}{2} \times \text{base} \times \text{height} \]This formula arises because the area of a triangle is essentially half the area of a parallelogram with the same base and height. To apply this formula, simply multiply the base by the height, and then divide the result by two.

• Base: The length of one side of the triangle.
• Height: The perpendicular distance from the base to the opposite vertex.

Make sure to plug in the correct values for base and height, and calculate accordingly for an accurate area measurement.

Applying Mathematical Formulas

Applying mathematical formulas efficiently requires understanding not just the formula itself, but also recognizing the context in which it is used. Let's apply the triangle area formula to find the area of the triangle with a base of 12 units and a height of 9 units. Substituting the given values into the formula, we have:\[A = \frac{1}{2} \times 12 \times 9\]Calculate:

• First, multiply 12 and 9 to get 108.
• Then, divide 108 by 2 to get 54 square units.

Understanding this step-by-step process helps reinforce how formulas function and the importance of plugging in values accurately. The final result, 54 square units, confirms the process was executed correctly.