A 1 2h B1 B2 Solve For B1

Understanding How to Solvefor b₁ in the Trapezoid Area Formula

The ability to solve for b₁ appears frequently when working with the area of a trapezoid, a shape defined by two parallel bases (b₁ and b₂) and a height (h). The standard formula for the area (A) of a trapezoid is

[A = \frac{1}{2},h,(b_1 + b_2) ]

Re‑arranging this equation to isolate b₁ is a straightforward algebraic process, yet it is a foundational skill that supports more complex geometry, physics, and engineering problems. In the sections below we break down each step, provide clear examples, highlight common pitfalls, and offer practice opportunities so you can confidently solve for b₁ in any context.

1. The Trapezoid Area Formula: A Quick Review

Before diving into the algebra, it helps to recall what each symbol represents:

The formula states that the area equals half the product of the height and the sum of the two bases. Geometrically, this reflects the fact that a trapezoid can be thought of as a rectangle plus two right triangles, or as the average of the bases multiplied by the height.

2. Step‑by‑Step Procedure to Solve for b₁

To isolate b₁, we treat the area formula as an equation and apply inverse operations in the reverse order of operations (PEMDAS/BODMAS). The process is:

1. Eliminate the fraction – Multiply both sides by 2 to remove the (\frac{1}{2}) factor.
2. Divide by the height – Cancel h by dividing both sides by h.
3. Subtract b₂ – Move b₂ to the opposite side to leave b₁ alone.

Written algebraically:

[ \begin{aligned} A &= \frac{1}{2},h,(b_1 + b_2) \ 2A &= h,(b_1 + b_2) \qquad\text{(multiply both sides by 2)}\ \frac{2A}{h} &= b_1 + b_2 \qquad\text{(divide both sides by h)}\ b_1 &= \frac{2A}{h} - b_2 \qquad\text{(subtract b₂ from both sides)} \end{aligned} ]

The final expression,

[ \boxed{b_1 = \frac{2A}{h} - b_2} ]

is the direct method to solve for b₁ whenever you know the area, height, and the other base.

3. Worked Examples ### Example 1: Basic Numerical Problem

A trapezoid has an area of 84 cm², a height of 7 cm, and one base measuring 9 cm. Find the length of the unknown base b₁.

Solution

[ \begin{aligned} b_1 &= \frac{2A}{h} - b_2 \ &= \frac{2 \times 84}{7} - 9 \ &= \frac{168}{7} - 9 \ &= 24 - 9 \ &= 15\text{ cm} \end{aligned} ]

Thus, b₁ = 15 cm.

Example 2: Using Decimal Values

A garden bed shaped like a trapezoid covers 45.5 m², has a height of 5 m, and the longer base (b₂) is 12 m. Determine the shorter base b₁.

Solution

[ \begin{aligned} b_1 &= \frac{2 \times 45.5}{5} - 12 \ &= \frac{91}{5} - 12 \ &= 18.2 - 12 \ &= 6.2\text{ m} \end{aligned} ]

The shorter base measures 6.2 m.

Example 3: Solving for b₁ When Area Is Expressed Algebraically

Suppose the area of a trapezoid is given by (A = 3h^2 + 2h), the height is (h), and the known base is (b_2 = h + 4). Find b₁ in terms of h.

Solution

[ \begin{aligned} b_1 &= \frac{2A}{h} - b_2 \ &= \frac{2(3h^2 + 2h)}{h} - (h + 4) \ &= \frac{6h^2 + 4h}{h} - h - 4 \ &= (6h + 4) - h - 4 \ &= 5h \end{aligned} ]

Hence, b₁ = 5h. This shows how the same algebraic steps work even when the quantities are expressions rather than fixed numbers.

Example 4: A Real-World Application – Roofing

A sloped roof of a house is shaped like a trapezoid. The area of the roof is 180 square feet, the height of the roof is 10 feet, and the longer side (b₂) is 25 feet. What is the length of the shorter side (b₁)?

Solution

We'll apply the formula we've learned:

[ b_1 = \frac{2A}{h} - b_2 ]

Substituting the given values:

[ b_1 = \frac{2 \times 180}{10} - 25 ]

[ b_1 = \frac{360}{10} - 25 ]

[ b_1 = 36 - 25 ]

[ b_1 = 11 \text{ feet} ]

Therefore, the shorter side of the roof is 11 feet. This example demonstrates the practical application of trapezoid area formulas in various scenarios, from architectural design to engineering calculations.

Conclusion

The formula for the area of a trapezoid, (A = \frac{1}{2}(b_1 + b_2)h), provides a powerful tool for calculating the area of this geometric shape. By understanding the relationship between the bases and the height, and by applying the algebraic steps to solve for an unknown base, we can determine the dimensions of trapezoids in a variety of situations. The examples presented illustrate the versatility of this formula, ranging from simple numerical problems to more complex scenarios involving algebraic expressions and real-world applications. Mastering this formula not only enhances our understanding of geometry but also equips us with practical skills applicable to fields like architecture, engineering, and even everyday problem-solving. The ability to calculate the area of a trapezoid is a fundamental skill with wide-ranging importance in mathematics and beyond.