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Chapter 1: Problem 48

Geometry. The area of a parallelogram is \(84 \mathrm{cm}^{2}\) The height of the figure is \(7 \mathrm{cm} .\) How long is the base?

Short Answer

Expert verified
The base is 12 cm.

Step by step solution

01

- Understand the Problem

The problem provides the area of a parallelogram and the height but asks for the base's length. The formula for the area of a parallelogram is needed.
02

- Write the Formula for Area

Recall the formula: \[ \text{Area} = \text{base} \times \text{height} \]
03

- Substitute the Given Values

Substitute the given values into the formula: \[ 84 = \text{base} \times 7 \]
04

- Solve for the Base

Isolate the base by dividing both sides of the equation by the height (7 cm): \[ \text{base} = \frac{84}{7} \]
05

- Calculate the Base

Perform the division: \[ \text{base} = 12 \text{ cm} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parallelogram
A parallelogram is a four-sided shape where each opposite side is parallel and equal in length. This means that if you look at a parallelogram, you'll see two pairs of opposite sides running alongside each other, never intersecting no matter how far they extend. A classic example is a slanted rectangle or a diamond-shaped figure. Important characteristics include:
  • Opposite sides are equal in length.
  • Opposite angles are equal in measure.
  • The diagonals bisect each other but are not necessarily equal.
Understanding these basic properties will help in solving various geometry problems involving parallelograms.
area formula
The area of a parallelogram can be found using a simple formula: \[\text{Area} = \text{base} \times \text{height} \]This formula indicates that to find the area, you multiply the length of the base by the perpendicular height. In this formula:
  • The base (\text{b}) is any one of the sides of the parallelogram.
  • The height (\text{h}) is the perpendicular distance from the base to the opposite side.
It's crucial to use the height that is perpendicular to the base, not the length of any slanted sides. This fundamental formula is a starting point for numerous geometry problems involving parallelograms.
base calculation
Calculating the base of a parallelogram involves rearranging the area formula. If the area and height are given, we use these steps:
1. Start with the formula: \[\text{Area} = \text{base} \times \text{height} \]
2. Substitute the known values into the formula. For example, if the area is 84 cm虏 and the height is 7 cm: \[\text{84} = \text{base} \times 7 \]
3. Solve for the base by dividing both sides by the height: \[\text{base} = \frac{84}{7} \]
4. Perform the division to find the base: \[\text{base} = 12 \text{ cm} \]By following these steps, you can easily find the length of the base when the area and height are known.
height
The height of a parallelogram, essential for calculating its area, is the perpendicular distance from the base to the opposite side. It鈥檚 important to note that:
  • The height is always perpendicular to the base.
  • It does not need to connect to a vertex; it can be a vertical distance dropped from the base to its parallel side.
For example, if given the area (\text{A}) and the base (\text{b}), you can find the height (\text{h}) by rearranging the area formula as follows: \[\text{h} = \frac{\text{A}}{\text{b}} \] Knowing how to determine the height is crucial for solving questions about the perimeter, area, and properties of parallelograms.

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