Next Generation Advanced Algebra and Functions Study Guide for the ACCUPLACER Test

Geometry Concepts for Algebra I

Many techniques in algebra involve solving for an unknown value that is embedded in an equation. Naturally, these methods find application in geometry problems, especially equations relating a known angle measurement to the smaller angles of unknown measurement constituting that larger angle. Algebraic methods extend into any situation in which an equation or formula is used to solve for a variable. A solid understanding of algebra enables many approaches and solutions to geometry problems to be quickly articulated.

Creating Expressions

Creating algebraic equations built upon geometric concepts often requires an understanding of the formulaic relationship between geometric concepts. Occasionally this will involve an explicit formula, an area or a volume, for example, and one unknown value, maybe the total area or only a side length, for example.

Identifying the known values and unknown values in a problem, followed by the correct set up in relating those values, followed by the correct application of algebraic and arithmetic rules is the best method for solving problems of this type.

Area

Area is a two-dimensional measurement of the size of a figure; it is always in \(units^2\). It is useful to familiarize yourself with some common area formulas:

Square: \(A = s^2\), where \(s\) is side length

Rectangle: \(A = lw\), where \(l\) is the length and \(w\) is the width

Triangle: \(A = \frac{1}{2}bh\) where \(b\) is the base and \(h\) is the height

Circle: \(A = \pi r^2\), where \(r\) is the radius

Trapezoid: \(A = h \cdot \frac{(b_1 + b_2)}{2}\), where \(h\) is the height, \(b_1\) is a base, and \(b_2\) is a base

Perimeter

The perimeter of a figure is the total distance around the figure. In the case of a circle, this distance is called the circumference. There are formulas that can be used to directly compute the perimeter of basic shapes, but it is nearly equally easy to just add up the side lengths of the shape in question.

The circumference; however, requires a special formula:

\(C = 2\pi r = \pi d\), where \(r\) is the radius and \(d\) is twice the radius

Remember that perimeter is always one-dimensional (\(units^1\)).

Volume

The volume of an object is the three-dimensional space that it occupies. Consequently, the units of any volume calculation must always be cubed (\(units^3\)).

Some common volume formulas with which you should be familiar are:

Cube: \(V = s^3\), where \(s\) is the side length

Rectangular prism: \(V = lwh\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height

Sphere: \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius

Cylinder: \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height

Cone: \(V = \frac{1}{3} \pi r^2 h\), where \(r\) is the radius and \(h\) is the height

Pyramid: \(V = \frac{1}{3} lwh\), where \(l\) and \(w\) are the length and width of the base, and \(h\) is the height

The Distance Formula

Given the location of two points on a coordinate plane, the distance between the points can be computed. The formula can be derived from the Pythagorean theorem, which is described in the next section:

\(d = \sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}\), where \(d\) is the distance and \((x_1, y_1)\) and \((x_2, y_2)\) are two points. The diagram shows the rationale underlying the formula:

Notice that we have generated a right triangle from the hypotenuse produced by the segment connecting points \(A\) and \(B\). The legs of the right triangle are formed by the distance between the \(x\) values and the distance between the \(y\) values.

The Pythagorean Theorem

The Pythagorean theorem describes the relationship between the legs of a right triangle and its hypotenuse.

The Pythagorean theorem states:

\(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse (longest side) of a right triangle.

Notice that \(c\) can be solved by evaluating the square root of both sides:

\(c = \sqrt{a^2 + b^2}\).

The similarity between this formula and the distance formula covered in the previous section should be readily apparent when the legs \(a\) and \(b\) are recognized as the difference between the \(x\) values and the difference between the \(y\) values, respectively.

Geometry Concepts for Algebra II

Geometry concepts extend beyond Algebra I. Commonly, Algebra II will introduce more advanced geometry concepts that were not covered or fully explored in Algebra I. These concepts relate to the volumes of curved objects, the equations governing conic sections, and theorems connecting the angle measurements inside of circles made by chords.

Volume of Non-Prism Objects

Non-prism objects are those containing non-flat sides. Spheres, cylinders, and cones represent the most common non-prism objects. Each has a formula for determining its volume.

These are the formulas:

Sphere: \(\frac{4}{3}\pi r^3\), where \(r\) is the radius

Cylinder: \(\pi r^2 h\), where \(r\) is the radius and \(h\) is the height

Cone: \(\frac{1}{3} \pi r^2 h\), where \(r\) is the radius and \(h\) is the height

In cases where you are given a shape that is a combination of other shapes, a rectangular prism that has a hemisphere attached to one end, for example, the total volume is the sum of the volumes of the contributing shapes.

Intersecting Line Theorems

It is useful to learn the following relationships among the chords and angles inside of circles.

• The measure of a central angle is the same as the measure of the arc it creates:

• If an angle is inscribed inside of a circle, the measure of the inscribed angle is half the measurement of the arc:

• If an angle is formed between a chord and a line tangent to the circle that intersects the chord, the angle measurement is half the measurement of the arc length formed:

• If an angle is formed by the intersection of two chords, the angle measurement is half the sum of the arc length measurements formed by the chords:

• If an angle is formed by two tangent lines, by two secant lines, or by a secant line and a tangent line intersecting outside a circle, the measure of the angle is half the difference between the major and minor arc measurements:

Circle Equations

There are two forms of the equation for a circle with which you should be familiar. The standard form of a circle is defined as:

\((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle, and r is the radius. Notice that when the origin is the center of the circle and the radius is \(1\), the equation reduces to:

\(x^2 + y^2 = 1\).

The general form of the equation of a circle is:

\(x^2 + y^2 + Ax + By + C = 0\), where A, B, and C are constants

To identify the center and radius of a circle given in general form, it is necessary to utilize the method of completing the square to convert the equation into the standard form. Once converted, the center and radius are readily apparent.