Question

Find the length of median \(\overline{A B}\) in trapezoid \(J K L M\) if \(J M=14\) inches and \(K L=21\) inches. (Lesson 8-5)

Short answer

The length of the median \( \overline{AB} \) is 17.5 inches.

Step-by-step solution

Understanding the Problem

We are given a trapezoid \( J K L M \) with bases \( JM = 14 \) inches and \( KL = 21 \) inches. We need to find the length of the median \( \overline{AB} \). According to the formula, the median length in a trapezoid is the average of the two bases.

Apply the Median Formula

The formula for the median \( \overline{AB} \) in a trapezoid is given by:\[\overline{AB} = \frac{JM + KL}{2}\]

Substitute the Values

Substitute the values of the bases \( JM \) and \( KL \) into the median formula:\[\overline{AB} = \frac{14 + 21}{2}\]

Calculate the Median Length

Perform the calculation:\[\overline{AB} = \frac{35}{2} = 17.5\]Thus, the length of the median \( \overline{AB} \) is 17.5 inches.

Key concepts

median of a trapezoid

The median of a trapezoid is a line segment that connects the midpoints of its non-parallel sides, sometimes known as the legs. This segment is unique because it holds a critical geometric property - it is parallel to the bases of the trapezoid. The role of the median is significant as it essentially helps in simplifying computations involving the trapezoid’s vertices and boundaries.
To find the length of the median, you simply take the average of the lengths of the two bases. If the bases are known, you can use the formula:\[\text{Median length} = \frac{\text{Base}_1 + \text{Base}_2}{2}\]This formula is useful because it not only gives an understanding of the median's position relative to the whole trapezoid but also provides a quick way to gauge the geometry's dimensions. Understanding this allows you to solve various problems involving the median of a trapezoid.

trapezoid bases

A trapezoid is a quadrilateral with at least one pair of parallel sides. These parallel sides are referred to as the trapezoid’s bases. In the context of geometry, having parallel sides makes it easy to explore the properties of trapezoids, such as correctness in calculations involving the area and median.
To distinguish between the two bases:- **Base 1 (denoted as \( JM \) in our problem):** The side of the trapezoid that measures 14 inches.- **Base 2 (denoted as \( KL \)):** The other parallel side that measures 21 inches.The bases act as reference lines for calculating distances and angles, and crucial to understand their measurements when dealing with geometry problems like solving for the median.

geometry formulas

In geometry, formulas are indispensable for calculating various attributes of shapes, like areas, perimeters, and segment lengths. They serve as tools that simplify what might otherwise be complex spatial relationships.
When it comes to a trapezoid, key formulas include:- **Area Formula:** Used to find the complete space inside the trapezoid, calculated as: \[ \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} \] This requires both the height and the lengths of the bases.- **Median Length Formula:** As shown previously, this is: \[ \text{Median length} = \frac{\text{Base}_1 + \text{Base}_2}{2} \] It simplifies finding the middle distance between two parallel lines, enhancing the understanding of trapezoidal shapes.With these formulas at your disposal, you can approach geometry problems with clarity and confidence, unraveling even the most challenging questions with ease.