Mathematics Study Guide for the ACT

Geometry: Part 1

Geometry is the study of shapes and their sizes and positions in space. It also includes the study of points, lines, and angles. We divide geometry into two types: plane geometry and solid geometry.

Plane geometry deals with \(2\)D shapes such as triangles, squares, and circles. Solid geometry is the study of \(3\)D shapes such as cubes, spheres, and pyramids. The concepts of congruence, similarity, and symmetry are part of geometry as well.

Lines

One of the most fundamental concepts of geometry is the line. A line is a straight path that extends infinitely in both directions. It is made up of an infinite number of points. A line has no thickness or width. We can name a line using any two points on the line.

A line segment is a portion of a line that is not infinite; it has a definite length. A line segment can be named using its two endpoints. Line segments are used to represent the shortest distance between two points.

A ray is also a portion of a line, but it has one endpoint and the other end extends infinitely. Rays are named using their endpoint and another point on the ray.

The symbols for a line, line segment, and ray are \(\overleftrightarrow{AB}, \overline{PQ},\text{ and } \overrightarrow{MN}\), respectively.

Finding Midpoint of Line Segments

The midpoint of a line segment is the point that divides the line segment into two congruent segments. (Note: Any two geometric figures that are exactly the same size and shape are congruent.) If one endpoint of a line has coordinates \((x_1,y_1)\) and the other endpoint has coordinates \((x_2,y_2)\), then the midpoint, \(m\), of the line segment is given by the formula:

\[m = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

This gives us the average of the two coordinates, \(x\) and \(y\), respectively.

Parallel Lines

Parallel lines are lines that never intersect. They maintain the same distance between themselves at all points. If two lines on a graph have the same slope, they are parallel.

Parallel lines can be used to find angle measures in a variety of ways. If a transversal cuts two parallel lines, the corresponding angles are congruent. They are angles that are in the same position relative to the transversal. Look at the figure below. It shows two parallel lines, \(m\) and \(n\), cut by a transversal, \(t\):

\(\angle A\) and \(\angle B\) are corresponding angles and thus are equal. Also, \(\angle C\) and \(\angle D\) are corresponding angles. They are congruent as well.

Finding Perimeter

The perimeter of a polygon is the sum of the lengths of all its sides. To find the perimeter of any polygon, simply add the length of all the sides. There are certain formulas for finding the perimeter of common shapes.

• square—\(4s\), where \(s\) is the side length

• rectangle—\(2l + 2w\), where \(l\) is the length and \(w\) is the width

• triangle—\(a+b+c\), where \(a\), \(b\), and \(c\) are the lengths of the three sides of the triangle

• circle—\(2\pi r\), where \(r\) is the radius of the circle

If you need to find the perimeter of an irregular figure, just add all the sides. Sometimes, you need to find the perimeter of composite figures. A composite figure is a \(2\)D shape that is formed by combining multiple basic \(2\)D shapes, such as triangles, circles, or squares. Usually, some sides aren’t given, you need to find the lengths of those sides and then add all the side lengths to find the perimeter. Look at the composite figure below:

First, we need to find the length of the blue side. That is \(12 - 8 = 4\). The length of the red side is \(15 - 5 = 10\). Now we know the lengths of all the sides. Thus, the perimeter of the composite figure shown is \(12 + 5 + 4 + 10 + 8 + 15 = 54\).

Finding Area

The area of a polygon is the amount of space enclosed within the boundaries of a shape. To find the area of basic polygons, such as triangles, rectangles, squares, and circles, use their respective formulas:

• square—\(s^2\), where \(s\) is the side length

• rectangle—\(lw\), where \(l\) is the length and \(w\) is the width

• triangle—\(\frac{1}{2}bh\), where \(b\) is the length of the base and \(h\) is the height

• circle—\(\pi r^{2}\), where \(r\) is the radius of the circle

To find the area of irregular polygons, divide them into smaller basic shapes and find the area of each shape. Then, add up the area of the smaller polygons to get the area of the larger shape.

Finding Volume

The volume is the amount of space occupied by a \(\boldsymbol{3}\)D object. To find the volume of common \(3\)D shapes, use the formulas shown below:

• cube—\(s^3\), where \(s\) is the side length

• rectangular prism—\(lwh\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height

• cylinder—\(\pi r^{2} h\), where \(r\) is the radius of the base and \(h\) is the height

• cone—\(\frac{1}{3} \pi r^{2} h\), where \(r\) is the radius of the base and \(h\) is the height

• sphere—\(\frac{4}{3} \pi r^{3}\), where \(r\) is the radius

Finding Surface Area

The surface area of a \(3\)D shape is the sum of the areas of all its faces. Sometimes, you will be given the volume of a shape and some more information, and then you will be asked to find the surface area. To do so, equate the given volume to the formula for that shape and find the unknown. Then, use all the known values to find the surface area of the figure.

Symmetry

Symmetry occurs when a line can be drawn through a shape or pattern so that it reveals two mirror images. Symmetry helps us find missing side lengths or angle measures in shapes by looking for patterns and similarities between different parts of the shape.

Below, the shape on the left is symmetrical, and its line of symmetry is shown. There is no line of symmetry in the shape on the right.