# Circles, Angles, Triangles, and Transformations

## Circles

• Circle: a set of points in a plane that are the same distance from a point called the center

• Chord: a line segment connecting any two points on a circle

• A chord may or may not go through the center of a circle

• Diameter: a chord that goes through the center of a circle

• Longest chord of a circle

• Radius: a line segment joining the center of a circle to any point on the circle

• The length of the diameter is twice the length of the radius

• Circumference: the distance around or “perimeter” of a circle

• Circumference approximately equals 3 * diameter or 6 * radius ### Example

If the radius of a circle is 5 inches, how long is the diameter?

The diameter of a circle is twice the diameter, so 2 * 5 inches = 10 inches.

## Angles

An angle is made up of two rays that share a vertex. Intersecting lines and line segments can also form angles.

Angles are measured in degrees (°). An angle that is open wider has a greater number of degrees. (For example, in the picture, the angle on the right has a greater number of degrees.)

A degree is 1/360 of a rotation of a full circle. There are 360° in a circle.  Types of angles: right, acute, obtuse, straight

• A right angle measures exactly 90°.

• An acute angle measures greater than 0° but less than 90°.

• An obtuse angle measures greater than 90° but less than 180°.

• A straight angle measures exactly 180°.

You should be able to roughly tell the different types of angles just by looking at them.

### Example

Angle C is the sum of angles A and B. Angle A is 72° and angle C is 162°. What is the measure of angle B and what type of angle is it?

We can write an equation for this problem.

A + B = C

Then we can plug in values.

72° + B = 162°

B = 90°

Since the measure of angle B is equal to 90°, it is a right angle.

## Triangles

Congruent: this word means that two things are the same; for example, two side lengths, two angles, two shapes

Congruent sides are marked with the same number of hatch marks on each congruent side.

IMPORTANT:

The sum of the interior angles of a triangle always adds up to 180°.

Classification of triangles:

- By angles:

• A right triangle has one right angle

• An obtuse triangle has one obtuse angle

• An acute triangle has three acute angles

- By side length:

• A scalene triangle has no congruent sides

• An isosceles triangle has at least two congruent sides

• An equilateral triangle is a type of isosceles triangle that has 3 congruent sides and angles, and each angle measures 60°  ### Example

Categorize the triangle on the left.

By sides: It is an isosceles triangle because there are two congruent sides (we know because of the hatch marks).

By angles: All the angles are less than 90°, so it is an acute triangle.

The triangle is an acute isosceles triangle.

### Example 2

What is the measure of angle C in the triangle on the right?

We know that the sum of the interior angles of a triangle is 180°. Since we know two of the angles, we can write an equation.

30° + 37° + C = 180°, so C = 113°. ## Transformations

A transformation of a figure changes the size, shape, or position of the figure to a new figure.

Congruent figures have the same size and shape and the orientation of shapes doesn't affect whether or not they're congruent.

Types of transformations:

• A translation (shift) is a transformation in which every point on the shape is moved the same distance in the same direction.

• A reflection (flip) is a transformation in which the shape is reflected over the line of reflection. All corresponding points in the original and new images are the same distance from the line of reflection.

• A rotation (spin) is a transformation in which the shape is rotated about a point called the center of rotation. The center of rotation may or may not be on the original image.

The resulting figure of a translation, reflection, or rotation is always congruent to the original figure. 