# Page 3 Geometry Study Guide for the Math Basics

### Circles

A circle is defined as the set of all points equidistant (the same distance) from a point called the center. That distance from the center is called the radius.

The distance across a circle through the center is called the diameter. Since the diameter is made up of two radii (the plural of radius), then $d=2r$.

The perimeter of a circle is called the circumference. A very important mathematical constant called Pi ($\pi$) is defined as the ratio of the circumference to the diameter ($\pi = \frac{C}{d}$). This number is the same for every circle and it turns out to be the irrational number $\pi = 3.14159…$. Usually, we rearrange the formula to find the circumference: $C=\pi \cdot d$ or, because $d=2r$, $C = 2 \pi r$.

The area of the circle is found using the formula $A = \pi \cdot r^2$.

For example, if a circle has a 3 cm radius, like this one, then:

and

Sometimes we use the decimal 3.14 as an approximation for $\pi$, so:

and

### Analyzing Figures

Some figures look quite similar to others, and some look like they might be able to be copied, twisted, and/or flipped to look like themselves.

#### Congruent

If two sides have the same length, we say they are congruent ($\cong$).
Likewise, if two angles have the same measure, they are congruent.

Two polygons are congruent if their corresponding sides and angles are congruent. Look at these two rectangles.

Within them, $\angle{A}$ corresponds to $\angle{E}$, $\angle{B}$ corresponds to $\angle{F}$, and so on. Likewise, $\overline{AB}$ corresponds to $\overline{EF}$, $\overline{BC}$ corresponds to $\overline{FG}$, and so on. All of the following are true:

So, we can say that $ABCD \cong EFGH$.

#### Similar

If two polygons have the same number of sides and each corresponding angle is congruent, we say they are similar. We use the symbol $\sim$ to show similarity. Look at the two figures.

In this case $\angle{A} \cong \angle{D}$, $\angle{B} \cong \angle{E}$, and $\angle{C} \cong \angle{F}$, so we say $\triangle ABC \sim \triangle DEF$.

For similar polygons, the sides are proportional. For these triangles, that means $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$.

#### Symmetry

A figure has symmetry if it can be folded or rotated onto itself. There are two main types of symmetry.

Line Symmetry (or fold symmetry) means a line exists where you can fold the figure exactly onto itself. This isosceles triangle has one line of symmetry.

This regular hexagon has 6 lines of symmetry.

A figure has rotational symmetry if you can rotate the figure less than $360^\circ$ onto itself. This equilateral triangle can be rotated $120^\circ$ clockwise (or counterclockwise) onto itself.

### Transformations

Figures can be manipulated by flipping, sliding, turning, or stretching. These are called transformations. If a transformation is performed, we call the initial figure the preimage and the final figure the image.

#### Reflection

A reflection is the result of flipping a figure over a line. In this case, the preimage $\triangle{ABC}$ has been reflected over the y-axis onto the image $\triangle{A’B’C’}$.

Note, it is common practice to label the vertices of the image as the primes of the vertices of the preimage. The point $A$ has been reflected onto $A’$ (A prime), for example.

#### Translation

A translation is the result of sliding a figure. In this case, the preimage $\triangle{P}$ has been translated up and to the right onto its image, $\triangle{P’}$.

#### Rotation

A rotation is the result of turning an object around a point called the center of rotation. In this case, trapezoid $H$ has been rotated $90^\circ$ clockwise about the origin onto trapezoid $H’$.

#### Dilation

A dilation occurs when a figure is stretched or compressed by an amount called the scale factor. In this case, the hexagon has been dilated by a scale factor of $1.5$, meaning each side of the image is one and a half times bigger than its corresponding side of the preimage.