Math Study Guide for the SHSAT

Geometry

Geometry is the branch of mathematics that deals with the study of shapes, sizes, properties, and dimensions of objects and spaces. It encompasses various concepts, including points, lines, angles, surfaces, solids, and their relationships.

Geometry plays a critical role in understanding the physical world around us, from the arrangement of molecules to the structure of buildings and the layout of cities. It provides essential tools for problem-solving, visualization, and reasoning and is vital in a wide number of career fields, including engineering, architecture, graphic design, and many others.

Polygons and 3-D Solids

A polygon is a closed two-dimensional (\(2\)-D) figure made up of line segments, often called the sides of the polygon. For example, squares and rectangles are polygons. There are two standard measurements for polygons: perimeter and area. The perimeter is the sum of the lengths of the sides of the polygon. The area is the measurement of the area enclosed by the polygon.

There are some common formulae for the perimeter of special polygons. For example, if the length of a rectangle is \(\ell\) and the width of the rectangle is \(w\), then the perimeter of the rectangle has the form \(2 \ell + 2w\) and the area of the rectangle is \(\ell \times w\).

A special class of polygons are called regular polygons. These are polygons where each side length is the same. For example, an equilateral triangle and a square are regular polygons. For any regular polygon with \(p\) sides, the sum of every interior angle has the same measure, namely \(180 \times (p - 2)\) degrees.

The most important measurement for a \(3\)-D solid is volume. The volume of a \(3\)-D solid is the measurement of the space within the solid. For example, a rectangular prism with length \(\ell\), width \(w\), and height \(h\) has volume \(\ell \times w \times h\).

Scale Drawings

A scale drawing of an object either enlarges or reduces the size of the object, while retaining its proportions. Enlargement alters the dimensions of an object by multiplying each measurement by a scale factor to make it larger, while reduction uses a scale factor to make its measurements smaller.

Typically, the scale of a drawing is expressed as a ratio. For example, \(1\text{:}5\) means that every unit on the drawing represents five units in the real physical space This ratio is often denoted with units, such as \(1\text{ cm}\) to \(5\text {m}\), which would mean that each centimeter on the diagram corresponds to five meters in the real world.

To correctly interpret and create scale drawings, it’s essential to know both the scale factor and the actual lengths of the object being represented.

In the example below, the left figure could be thought of as the scaled drawing of the larger figure on the right. In this example, it can be seen that the scale is \(1 \text{:} 5\).

Translating Scale Drawings

Translating scale drawings involves converting the scaled-down or scaled-up representation back into actual proportions. This process requires determining the relationship between the measurements in the drawing and the corresponding measurements in real life. By understanding the scale factor used in the drawing, one can accurately interpret the dimensions and features represented.

Creating Equivalent Drawings

Creating equivalent drawings involves producing scale drawings with different scaling factors while maintaining the proportional relationships of the original object or space. This process allows for flexibility in representation, as different scales may be more suitable for specific purposes or contexts. By adjusting the scale factor, one can emphasize different aspects of the object or space, highlight details, or provide a broader overview.

Triangles

Triangles are a specific type of polygon with only three sides. There are a number of different types of triangles that are defined by their angles and side lengths. The relationship between the angles and side lengths helps us compare one triangle to another.

Similar Triangles

When one triangle is a scaled version of the other one, these are called similar triangles. Given two triangles, there are a few ways to tell if they are similar. These are called the triangle similarity theorems. They are as follows:

• angle-angle (AA)—The AA similarity theorem states that if two triangles have two angles that are equal to each other, then the triangles are similar.

• side-angle-side (SAS)—The SAS similarity theorem states that if two triangles have two proportional sides with the angle between these two sides equal, then the triangles are similar.

• side-side-side (SSS)—The SSS similarity theorem states that all sides of a triangle are proportional to all sides of another triangle, then these two triangles are similar.

Note: There is no angle-side-side similarity theorem.

In the example above, we are able to conclude that these two triangles are similar using either the SAS (side-angle-side) similarity theorem or the SSS (side-side-side) similarity theorem.

Triangle Properties

Triangles have two properties than can be useful at times:

• triangle inequality theorem—The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

• angle sum property—The angle sum property, also known as the triangle sum theorem, states that the sum of the interior angles of any triangle is always equal to \(180\) degrees.

Observe that when the triangle is regular, the angle sum property for triangles can be deduced from the angle sum property for regular polygons as mentioned above.

When constructing triangles with varying inputs, different outcomes can arise: a unique triangle, multiple triangles, or no triangle at all.

• For a unique triangle, the given information must uniquely determine the triangle’s sides and angles. This occurs when specific combinations of side lengths and angle measures lead to only one possible triangle configuration.

• Conversely, multiple triangles can be formed when the given information allows for more than one valid combination of sides and angles. In such cases, careful consideration and exploration are required to identify and construct all possible triangles.

• It’s also possible that the provided information does not allow for the construction of a valid triangle at all. This scenario occurs when the given side lengths or angle measures cannot satisfy the triangle inequality theorem or the angle sum property.

Recognizing these differing results is crucial for effectively solving geometric problems and understanding the constraints of triangle construction.

Circles

Circles are fundamental geometric shapes defined by a set of points in a plane that are equidistant from a fixed center point. Working with circles requires understanding key concepts, such as circumference and area. For the SHSAT, you’ll need to know the following formulas and be able to apply them.

Circumference

The circumference of a circle is the distance around its outer boundary. It can be calculated using this formula:

\[C=2 \pi r\]

where \(C\) represents the circumference, \(\pi\) is the mathematical constant pi (approximately equal to \(3.14159\)), and \(r\) is the radius of the circle. The radius is the distance from the center point of the circle to any point on its outer boundary.

Area

The area of a circle is the measure of the region enclosed by its boundary. It can be calculated using this formula:

\[A=\pi r^2\]

where \(A\) represents the area, \(\pi\) is the mathematical constant, and \(r\) is the radius of the circle.

Slicing Three-Dimensional Shapes

Slicing three-dimensional shapes to create two-dimensional shapes is a helpful concept in geometry, allowing us to visualize and analyze the cross-sectional properties of various solids. When slicing these \(3\)-D shapes, the resulting \(2\)-D shapes depend on the orientation and location of the slice relative to the shape’s geometry.

A question about this will state whether the slice is vertical or horizontal. The slices may form rectangles, triangles, or other polygons, providing insights into the internal structure and properties of the original \(3\)-D objects. Let’s explore this concept in the context of right rectangular prisms and right rectangular pyramids.

Right Rectangular Prisms

A right rectangular prism, also known as a rectangular solid or box, is a three-dimensional solid figure with six rectangular faces, where each face meets its adjacent faces at right angles. It has eight vertices and twelve edges.

The faces opposite each other are congruent rectangles. Right rectangular prisms are commonly encountered in everyday objects like boxes, bricks, and buildings.

If you were to take a horizontal slice of the right rectangular prism shown above, it would look like this two-dimensional figure:

Right Rectangular Pyramids

A right rectangular pyramid is a three-dimensional solid figure with a rectangular base and triangular faces that converge to a single vertex above the base. The base of the pyramid is a rectangle, and the height of the pyramid is the perpendicular distance from the base to the apex (top vertex).

One horizontal slice of the right rectangular pyramid shown above may look like this:

Understanding the slicing process enables us to analyze the relationships between \(3\)-D solids and their corresponding \(2\)-D representations.

Angle Relationships

Understanding angle relationships is necessary for solving geometric problems and finding unknown angles. Key concepts include supplementary and complementary angles, vertical angles, and adjacent angles.

Supplementary and Complementary Angles

Supplementary angles are pairs of angles whose measures add up to \(180\) degrees. Complementary angles, on the other hand, are pairs of angles whose measures add up to \(90\) degrees. These relationships can be expressed graphically as follows:

Supplementary angles:

\[\angle A+\angle B=180\]

Complementary angles:

\[\angle A+\angle B=90\]

Knowing these relationships allows us to solve problems involving angle measures by setting up equations to find unknown angles.

Adjacent Angles

Adjacent angles are defined as two angles with a common vertex and a common side. Depending on their arrangement, adjacent angles can either be complementary or supplementary. Complementary adjacent angles share a common vertex, side, and add up to \(90\) degrees, while supplementary adjacent angles share a common vertex or side and add up to \(180\) degrees.

Vertical Angles

Vertical angles are pairs of nonadjacent angles formed by intersecting lines. They are always congruent, meaning they have the same measure. If you consider the graphic below, this relationship can be expressed as:

\[\angle ∠1=\angle ∠3\] \[\angle ∠2=\angle ∠4\]

Problem-Solving

Geometry offers a wide array of problem-solving opportunities, from analyzing the properties of basic shapes to tackling complex scenarios involving three-dimensional figures. In this section, we’ll delve into three key areas: triangles and trapezoids, surface area calculations, and right rectangular prisms. Each domain presents unique challenges and requires the application of geometric principles and formulas to find solutions.

Triangles and Trapezoids

A trapezoid is any quadrilateral with at least one pair of parallel sides, which includes both rectangles and parallelograms.

Given trapezoid \(ABCD\) with parallel sides \(AB\) and \(CD\), where \(AB = 8\,\text{cm}\), \(CD = 12\,\text{cm}\), and the distance between the parallel sides (height) is \(6\,\text{cm}\), find the area of the trapezoid.

To find the area (\(A\)) for the trapezoid, we can use the following formula for the area of a trapezoid:

\[A= \frac{1}{2} \times s \times h\]

where \(s\) stands for the sum of the lengths of the parallel sides and \(h\) stands for the height of the trapezoid.

Substituting in the given values, we get:

\[​A=\frac {1}{2} \times (8+12) \times 6\] \[​=\frac {1}{2} \times 20 \times 6\] \[=\frac {1}{2} \times 120\] \[=60\]

Therefore, the area of the trapezoid \(ABCD\) is \(60\) square centimeters (\(\text{cm}^2\)).

Note: The United States and the United Kingdom have different definitions for the word “trapezoid.” The SHSAT uses the US definition, as described above.

Surface Area

The surface area of a solid is the sum of the area of all of its outside surfaces. During the SHSAT, you may need to work with the surface area of right prisms (which include cubes) and right pyramids that are made of triangles and trapezoids.

Consider a rectangular prism with dimensions \(6\,\text{cm}\) by \(4\,\text{cm}\) by \(8\,\text{cm}\). Find its total surface area.

The surface area (\(A\)) of a rectangular prism with a length \(l\), width \(w\), and height \(h\) is given by the following formula:

\[A=2lw+2lh+2wh\]

Substituting in the given values, we get:

\[A=2(6\times4)+2(6\times8)+2(4\times8)\] \[=2(24)+2(48)+2(32)\] \[=48+96+64\] \[=208\]

Therefore, the surface area of the rectangular prism is \(208\,\text{cm}^2\).

Right Rectangular Prisms

Problem-solving with right rectangular prisms often involves finding volume, surface area, or other properties of these solid figures. Let’s explore how to tackle such problems.

Suppose you have a box-shaped container that measures \(10\,\text{in}\) by \(6\,\text{in}\) by \(4\,\text{in}\). Determine its volume.

The volume (\(V\)) of a rectangular prism with length \(l\), width \(w\), and height \(h\) is given by the formula:

\[V=l\times w\times h\] \[=10 \times 6 \times 4\] \[=240\]

Therefore, the volume of the box-shaped container is \(240\) cubic inches (\(\,\text{in}^3\)).