Quantitative Reasoning Study Guide for the DAT

Quadrilaterals

Quadrilaterals are polygons with four sides and four vertices. Let’s explore different types of quadrilaterals and their unique properties.

A trapezoid is a quadrilateral with at least one pair of parallel sides. Its defining feature is the presence of exactly one pair of parallel sides. This shape is crucial in geometry, often appearing in real-world scenarios such as architectural designs and construction projects.

One notable property of trapezoids (and quadrilaterals in general) is that the sum of their interior angles always equals \(360^{\circ}\). When calculating the area of a trapezoid, one can utilize specific formulas based on the lengths of its parallel sides and height.

Parallelograms are quadrilaterals with opposite sides that are parallel and equal in length. Parallelograms possess symmetrical properties, such as having opposite angles that are equal and consecutive angles that are supplementary. Additionally, the diagonals of a parallelogram bisect each other, meaning they intersect at their midpoints, forming congruent segments.

A rhombus is a special type of parallelogram with four sides of equal length. This uniformity in side lengths distinguishes the rhombus from most other quadrilaterals. Their four angles, however, may or may not be of equal measure.

Rectangles are quadrilaterals with four right angles, making them a special case within the category of parallelograms. Rectangles exhibit symmetry in their opposite sides and diagonals, with the diagonals being congruent.

A square is a type of a rhombus and a rectangle, possessing all their respective properties. In addition, they have four congruent sides and four right angles, making them one of the most symmetrical shapes in geometry.

Measuring Figures

Measuring figures, particularly in terms of perimeter and area, is a geometry concept with significant relevance to standardized tests like the DAT exam. Let’s delve into this topic further.

Perimeter

Perimeter represents the total distance around the outside of a figure. For each type of quadrilateral, specific formulas exist for calculating perimeter based on their unique properties. For the five quadrilaterals we talked about, the perimeter formulas are shown below:

Area

Area represents the amount of space enclosed by a figure. Formulas for calculating area vary depending on the type of figure. For the above quadrilaterals, here are their area formulas:

Circles

Circles are foundational shapes in geometry, consisting of all points in a plane that are equidistant from a fixed point called the center. Mastery of circle properties and measurements helps to complete your understanding of basic geometry. Let’s explore circles further.

Parts and Terms

A circle is a closed two-dimensional shape that consists of points that are all the same distance from a center point. Thus, the center (or origin) of a circle is the fixed point from which all points on the circle are equidistant. The diameter of a circle is the longest chord (line) passing through the center. The radius is the distance from the center to any point on the circle. The length of the diameter is always twice that of the radius.

Measuring Circles

Measuring circles involves understanding various properties such as circumference, area, arc length, and sector area.

The circumference (\(C\)) of a circle is the distance around its boundary. It is the special word for the perimeter of a circle. It is calculated using this formula:

\[C = 2\pi r\]

where \(r\) is the radius of the circle and \(\pi\) is a mathematical constant roughly equivalent to \(3.14159\). For example, if a circle has a radius of \(10\) centimeters, the circumference of the circle would be \(C = 2\pi r = 2\pi (10) = 20\pi\) centimeters.

The area of a circle is the space enclosed by its boundary. It is calculated using this formula:

\[A = \pi r^{2}\]

where \(r\) is the radius of the circle. For example, if a circle has a radius of \(2\) centimeters, its area is going to be \(A = \pi (2)^{2} = 4\pi\) square centimeters. The area is always in square units.

An arc of a circle is a segment of its circumference. The length of an arc depends on the central angle it subtends. In this case, subtends means that the arc is formed by the points on the circle that are connected by a central angle. A central angle is an angle whose vertex is the center of the circle. The length of an arc is calculated using this formula:

\[l = \frac{\theta}{360} \times 2\pi r\]

where \(l\) is the arc length, \(\theta\) is the central angle in degrees, and \(r\) is the radius. If an arc of a circle, with radius \(3\) inches, subtends a central angle of \(60^{\circ}\), the arc length is calculated as follows:

\[l = \frac{60}{360} \times 2\pi (3)\] \[l = \frac{1}{6} \times 6\pi\] \[l = \pi \text{ in}\]

A sector of a circle is the region bounded by an arc and two radii. The area of a sector can be calculated using this formula:

\[A_{s} = \frac{\theta}{360} \times \pi r^{2}\]

where \(A_{s}\) is the area of the sector, \(\theta\) is the central angle in degrees, and \(r\) is the radius. For example, if a circle has a radius of \(6\) inches and the central angle of the sector is \(120^{\circ}\), the area of the sector is calculated as follows:

\[A_{s} = \frac{120}{360} \times \pi (6)^{2}\] \[A_{s} = \frac{1}{3} \times 36\pi\] \[A_{s} = 12\pi \text{ in}^2\]

Complex Figures

Complex figures are composite shapes composed of two or more simpler shapes. They frequently appear on standardized tests and may involve calculations of area or perimeter. These questions require knowledge of both the individual shapes and how they interact to form the complex figure. Consider the following shape:

Let’s find the area and perimeter of this complex shape. First, the perimeter is the length of the outer boundary of the figure. The bottom and top sides are \(10+3=13\) each. The left side is \(4+4 = 8\). The two slanted sides (to the right side of the figure) are the hypotenuses of two right triangles. So we can use the Pythagorean theorem to figure out the length of the slanted side (\(s\)):

\[s^2 = 4^2 + 3^2\] \[s^2 = 25\] \[s = \sqrt{25} = 5\]

So, the total length of the two slanted sides is \(5 + 5 = 10\).

Thus, the perimeter of the figure is:

\[13 + 13 + 8 + 5 + 5 = 44\text{ units}\]

To find the area, we dissect the complex shape into a rectangle and two right triangles. Then, we will add up the areas of the three parts to get the total area of the complex figure.

One side of the rectangle is \(10\) and the other side is \(8\). So, the area is \(10 \times 8 =80\). The base of the triangle is \(3\) and the height is \(4\). So, the area of the triangle is \(\frac{1}{2} \times 3 \times 4 = 6\). Since there are two congruent triangles, the total area of them is \(6 + 6 = 12\).

Thus, the area of the complex figure is:

\[80 + 12 = 92\text{ units}^2\]

Solid Geometry

Solids are three-dimensional (\(3\)-D) shapes that occupy space and have measurable attributes such as volume and surface area. Solid geometry is the branch of geometry concerned with the study of these three-dimensional figures. It involves analyzing their properties, including their faces, edges, and vertices, to understand their structure and calculate various quantities associated with them.

Parts of Solid Figures

In solid geometry, understanding the different parts of three-dimensional figures is essential for calculating their properties accurately. Two fundamental parts of solids are the base and height.

The base of a solid is the bottom face or surface upon which it rests or is constructed. It serves as a reference point for determining other measurements and attributes of the solid. For example, in a rectangular prism, the two congruent rectangles at the top and bottom are the bases. The height of a solid is the perpendicular distance between its base and its topmost point or face. It is a crucial dimension in determining the volume and surface area of the solid. For instance, in a cylinder, the height is the distance between its two circular bases.

Surface Area

Surface area refers to the total area of all the faces or surfaces of a solid figure. The general formula for calculating the surface area of three-dimensional shapes varies depending on the type of solid. If you know the formulas for finding the areas of various geometrical shapes, it will be easy for you to find the surface area of any \(3\)-D solid by adding up the areas of all its surfaces. Let’s do an example.

For the rectangular prism shown above, find its surface area.

The surface area is the sum of the areas of all faces of a solid. Each face here is a rectangle. Therefore, the area of the bottom is:

\[10 \times 2 = 20\text{ units}^2\]

The top has the same area.

The area of the right face is:

\[4 \times 2 = 8\text{ units}^2\]

The left face has the same area.

The area of the front face is:

\[10 \times 4 = 40\text{ units}^2\]

The back face has the same area.

Thus, the total surface area of the rectangular prism is \(20 + 20 + 8 + 8 + 40 + 40 = 136\text{ units}^2\).

Volume

Volume (\(V\)) is the measure of the amount of space occupied by a three-dimensional figure. As with surface area, the general formula for calculating the volume of solids also varies based on the type of solid. In general, the volume of a uniform \(3\)-D shape is the product of the area of its base and its height. Let’s do an example.

For the cylinder shown above, find the volume.

We need to find the area of the base, which is a circle, so:

\[A = \pi r^{2}\] \[A = \pi (2)^{2}\] \[A = 4\pi \text{ units}^2\]

To get the volume, we multiply the area of the base by the height:

\[V = 4\pi \times 5 = 20\pi \text{ units}^3\]

Note: Just as area is always given in square units, volume is always given in cubic units.

Spheres

A sphere is a perfectly symmetrical three-dimensional object in which every point on its surface is equidistant from its center.

Surface Area

The surface area (\(SA\)) of a sphere represents the total area of its outer surface. It is calculated using this formula:

\[SA = 4\pi r^{2}\]

where \(r\) represents the radius of the sphere. For example, if the radius of a sphere is \(3\) centimeters, its surface area will be \(SA = 4 \pi (3)^{2} = 36\pi \text{ cm}^3\).

Volume

The volume of a sphere is the measure of the space enclosed by its surface. It is calculated using this formula:

\[V = \frac{4}{3} \pi r^{3}\]

where \(r\) is the radius of the sphere. For the previously discussed sphere, the volume would be \(V = \frac{4}{3} \pi (3)^{3} = 36\pi \text{ cm}^3\).

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a branch of mathematics that combines elements of algebra and geometry. It deals with geometric figures by representing them using coordinate systems, usually the Cartesian coordinate system. This system assigns numerical values to each point in a plane, allowing geometric concepts such as distance, midpoint, slope, and equations of lines to be expressed algebraically.

Midpoints

Finding the midpoint of a line segment involves determining the point that lies exactly halfway between its endpoints. The midpoint (\(M\)) formula, derived from the average of the \(x\)-coordinates and \(y\)-coordinates of the endpoints, is:

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

If you have point \(A\) with coordinates \((2,4)\) and point \(B\) with coordinates \((-6,4)\), to find the midpoint of this line segment, find the average of the \(x\)-coordinates and \(y\)-coordinates, separately:

\[\left(\frac{2-6}{2}, \frac{4+4}{2}\right)\] \[= (-2, 4)\]

Distances

The distance between two points in a coordinate plane is the length of the straight line segment that connects them. The distance formula, derived from the Pythagorean theorem, is:

\[\sqrt{(y_2 - y_1)^{2} + (x_2 - x_1)^{2}}\]

For the two points \(A\) and \(B\) shown above, let’s calculate the distance of the line segment between the points:

\[\sqrt{(4-4)^{2} + (-6-2)^{2}}\] \[= \sqrt{(0)^2 + (-8)^{2}}\] \[= \sqrt{64}\] \[= 8\]

Slope

In terms of a line, the slope represents the measure of its steepness or incline. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line. The slope formula is:

\[m= \frac{y_2 -y_1}{x_2 - x_1}\]

The intercept of a line refers to the point where it intersects the \(y\)-axis. There are four types of slopes:

• positive slope
• negative slope
• undefined slope (vertical line)
• zero slope (horizontal line)

To understand these better, look at the figures below:

For the points \(A\) and \(B\) discussed above, let’s calculate the slope:

\[m = \frac{4-4}{-6-2}\] \[m = \frac{0}{-8}\] \[m = 0\]

Since the slope is \(0\), the line, when graphed, is a horizontal line.

Graphing Inequalities

Linear inequalities are mathematical statements that involve linear expressions and one of these comparison operators: \(<\), \(\leq\), \(>\), or \(\geq\). These are examples of linear inequalities: \(y < 2x + 5\), \(y \geq x - 2\), and \(y \leq 5x\).

To graph linear inequalities, follow these steps:

1. Graph the corresponding linear equation. If the inequality is strict (\(<\) or \(>\)), use a dotted line. If the inequality is not strict (\(\le\) or \(\ge\)), the line should be solid.

2. Choose a test point not on the line and substitute its coordinates into the inequality.

3. If the inequality is true for the test point, shade the region containing the test point; otherwise, shade the opposite region. Repeat for additional inequalities.

As an example, let’s graph the inequality \(y < 2x + 3\).

First, we graph the corresponding linear equation \(y = 2x + 3\). Note that the slope of this line is \(2\) and the \(y\)-intercept is \((0,3)\). Now, we choose a test point, for example \((0,0)\), and check the inequality:

\[0 \stackrel{?}{<} 2(0) + 3\] \[0 < 3\]

The inequality is true for this test point, so we shade the region that is below the line.