Problem 98 - Beginning and Intermediate Algebra

Question

A trapezoid is a four-sided figure with exactly one pair of parallel sides. The expression \(\frac{1}{2} h(B+b)\) represents its area, when \(B\) and \(b\) are the lengths of the two parallel sides and \(h\) is the height between these sides. Find the area if \(B=15\) inches, \(b=7\) inches, and \(h=5\) inches.

Step-by-Step Solution

Answer

The area of the trapezoid is 55 square inches.

Step 1: Identifying the Formula

The formula for the area of a trapezoid is given by \( \frac{1}{2} h (B + b) \). Here, \(h\) is the height, \(B\) is the length of one parallel side, and \(b\) is the length of the other parallel side.

Step 2: Substituting Known Values

Substitute the given values into the formula: \( B = 15 \) inches, \( b = 7 \) inches, and \( h = 5 \) inches. Thus, the expression becomes \( \frac{1}{2} \times 5 \times (15 + 7) \).

Step 3: Simplifying the Expression

First, calculate the sum of the parallel sides: \( 15 + 7 = 22 \). Then the expression becomes \( \frac{1}{2} \times 5 \times 22 \).

Step 4: Calculating the Area

Now, multiply the height by the sum of the bases: \( 5 \times 22 = 110 \). Finally, apply the factor of \( \frac{1}{2} \) to get the area: \( \frac{1}{2} \times 110 = 55 \). Therefore, the area of the trapezoid is 55 square inches.

Key Concepts

Geometry

Geometry is the study of shapes and their properties. It involves understanding various aspects such as angles, sides, and areas of different figures. In geometry, you often encounter various shapes like triangles, circles, rectangles, and trapezoids. This subject is essential because it helps us comprehend the spatial arrangement and dimensions of objects around us.
Trapezoids are one of the fascinating shapes studied in geometry. They have unique properties distinct from other quadrilaterals. Mastering these concepts provides a solid mathematical foundation that can be applied in everyday life and advanced mathematical studies. Learning geometry strengthens essential problem-solving skills.

Area Calculation

Calculating the area of a shape is a fundamental concept in geometry. The area refers to the amount of space enclosed within a shape. Knowing how to calculate areas is crucial for many applications, like determining the size of land plots or fabric needed for clothing.
For trapezoids, the area is calculated using a specific formula. This formula takes into account the lengths of the parallel sides and the height of the trapezoid. By plugging these values into the formula, you can find the trapezoid's area accurately and efficiently.

Trapezoid Properties

A trapezoid is a specific type of quadrilateral. The defining property of a trapezoid is having exactly one pair of parallel sides. These parallel sides are called the bases, while the other sides are non-parallel. The distance between the parallel sides, measured perpendicularly, is called the height.

• The sum of the internal angles in any quadrilateral, including a trapezoid, is always 360 degrees.
• Trapezoids can be classified further based on additional properties, such as being isosceles, where the non-parallel sides are equal in length.
• The area formula used for trapezoids is unique because it averages the lengths of the parallel sides before multiplying by the height.

Understanding these properties helps in recognizing and solving problems involving trapezoids accurately.

Mathematical Formulas

Mathematical formulas are essential tools in computations. They provide a standard way to perform calculations quickly and accurately. In the case of trapezoids, the area formula is expressed as:\[ A = \frac{1}{2} h (B + b) \]

• \( B \) and \( b \) represent the lengths of the parallel sides (bases).
• \( h \) is the perpendicular distance between the bases, known as the height.
• The expression \( (B + b) \) calculates the sum of the bases, which is then multiplied by \( h \) to find the area.
• The factor of \( \frac{1}{2} \) ensures the formula accounts for the trapezoidal shape by averaging the two bases.

Using these formulas makes the solution process systematic and understandable, allowing you to tackle various mathematical problems with ease.