Question

Algebra What is the base of a parallelogram with area 45 square yards and height 9 yards?

Short answer

The base of the parallelogram is 5 yards.

Step-by-step solution

Identify the formula for the area of a parallelogram

The area of a parallelogram is calculated using the formula: \( \text{Area} = \text{base} \times \text{height} \). In this problem, the given area is 45 square yards, and the height is 9 yards.

Set up the equation using the given values

Substitute the known values into the area formula. We replace the area with 45 and the height with 9, giving us the equation: \( 45 = \text{base} \times 9 \).

Solve for the base

To find the base, divide both sides of the equation by 9: \( \text{base} = \frac{45}{9} \). Simplifying this fraction gives us \( \text{base} = 5 \).

Conclusion

The base of the parallelogram is 5 yards, given the area and height provided.

Key concepts

Parallelogram Area Formula

A fundamental concept in geometry is finding the area of a parallelogram. The parallelogram area formula helps us calculate the space inside the shape. We use the equation:

• \( \text{Area} = \text{base} \times \text{height} \)

The base is the length of one side, while the height is the perpendicular distance from the base to the opposite side. It's important to note that the height must be perpendicular to the base. This formula allows us to understand how factors like the length of the base and the height impact the overall size of the parallelogram. To find one of these dimensions, we'll rearrange the formula depending on which values are given, demonstrating the versatility and practicality of algebraic manipulation. By practicing with this formula, we learn to connect visual geometry concepts with numerical calculations.

Algebra

Algebra is a key tool for solving geometric problems, like the one we have with the parallelogram. When given the area and one side measure, algebra helps us find the missing dimension. By translating the situation into an equation, we can systematically uncover unknowns. In our parallelogram example, we had:

• Area = 45 square yards
• Height = 9 yards

These were placed into the area formula as \( 45 = \text{base} \times 9 \). Using algebra, we solved for the base by isolating the variable:

• \( \text{base} = \frac{45}{9} = 5 \)

This showcases how algebra can simplify complex problems into manageable steps, showing that math is like a toolbox where each formula and operation serves a different purpose. Whether simplifying or solving, algebra allows for efficient problem-solving by logically structuring equations to find unknown quantities.

Measurement in Geometry

Measurement in geometry involves understanding various properties of shapes and how to calculate them. In our discussion on parallelograms, we've dealt with concepts such as area, base, and height. Each measurement gives us insight into the shape's structure and size.

• The area helps us understand the total surface the shape covers.
• The base is a linear measure of one side of the parallelogram.
• The height is the shortest distance from the base to the opposite side, measured perpendicularly.

Recognizing where measurements apply is crucial to interpreting and solving geometric problems accurately. By connecting these measurements through formulas, like the parallelogram area formula, we interpret these abstract values for practical use. Engaging with measurement actively sharpens analytical skills while bridging visual understanding of geometry with real-world applications.