Lesson 1: Measuring Segments

Property

Two segments are congruent if they have equal lengths: $\overline{AB} \cong \overline{CD}$ if and only if $AB = CD$. Congruent segments are marked with identical tick marks in geometric diagrams.

Examples

Property

In mathematics, the perimeter means the length of the boundary of a region. To find the perimeter of a straight-sided figure, measure the length of each side and add up the lengths of the sides.

Examples

• A rectangular garden is 10 meters long and 7 meters wide. Its perimeter is the sum of all its sides: $10 + 7 + 10 + 7 = 34$ meters.
• A triangular park has sides measuring 50 feet, 60 feet, and 75 feet. The perimeter is found by adding these lengths: $50 + 60 + 75 = 185$ feet.
• For a square window with one side measuring 2 feet, the perimeter is $4 \times 2 = 8$ feet, since all four sides are equal.

Explanation

Imagine you are an ant walking around the edge of a shape. The total distance you walk in one full circuit is its perimeter. It measures the length of the boundary, not the space inside.

Property

Perimeter problems use the total distance around a shape to form an equation. For a rectangle with length $L$ and width $W$, the perimeter is $P = 2L + 2W$. Some problems may involve fencing only some sides, such as $L + 2W$ for a yard against a house.

Examples

• The perimeter of a rectangular park is 100 meters. The length is 10 meters more than the width. The system is $\begin{cases} 2L + 2W = 100 \\ L = W + 10 \end{cases}$. The dimensions are $L=30$ m and $W=20$ m.

• A fence is built around three sides of a rectangular yard adjacent to a house, using 140 feet of fencing. The length is 20 feet more than twice the width. The system is $\begin{cases} L + 2W = 140 \\ L = 2W + 20 \end{cases}$. The dimensions are $L=68$ ft and $W=36$ ft.