Geometry

The portion of a line between, and including, two points is called a line segment. Line segments have a definite length that can be determined if enough information is provided.

Consider the line segment AC, which includes point B. If line segment AC measures 12, line segment AB measures 3x and line segment BC measures 4x, what is the length of AB?

Notice that because B lies along AC, the length of AB combined with the length of BC, yields the length of AC. As a result, an algebraic equation can be generated from the information provided:

AB + BC = AC

3x + 4x = 12

7x = 12

x = 12/7

Substitute the known value of x into the expression for AB:

3x = 3 * 12/7 = 36/7

The midpoint of a line segment is the point along the segment that divides the segment into two equal portions.

Consider segment XZ, which has midpoint Y. If XY measures 3x + 1 and YZ measures x + 2, what is the length of XZ?

Because Y is the midpoint, XY and YZ are of equal length. Set their lengths equal to each other to solve for the unknown x then substitute this value into the sum of XY and YZ to determine the length of XZ:

XY = YZ

3x + 1 = x + 2

2x = 1

x = 1/2

and

XY + YZ = XZ

3 * 1/2 + 1 + 1/2 + 2 = XZ

3/2 + 1/2 + 3 = XZ

5 = XZ

Here are some types of lines and their definitions:

Parallel lines are lines that never intersect because they have the same slope, but different x and y intercepts (if graphed on a coordinate plane). Parallel lines that intersect are the same line. Consider why this is.
Perpendicular lines are lines that intersect at a 90-degree angle. The slopes of perpendicular lines are negative reciprocals of each other. For example, if the slope of line a is 2, and line b is perpendicular to a, line b has a slope of −12.
A transversal is a line that passes through two parallel lines. Because parallel lines are essentially the same line, the angles formed by a transversal and one of the parallel lines are equivalent in measure to the corresponding angles formed on the other parallel line.

Angles

The intersection of lines, line segments, or rays generates angles. Acute angles are less than 90 degrees, right angles are exactly 90 degrees, and obtuse angles are greater than 90 degrees.

In the case of a transversal passing through two parallel lines, 4 pairs of corresponding angles are formed that have different relationships with each other. Each corresponding angle pair is congruent. Each of these angles also forms two linear pairs with the angles adjacent and a congruent vertical angle pair. The alternate exterior angles formed are congruent and the alternate interior angles formed are congruent, as well.

Straight angles measure 180 degrees. And a collection of angles around a point sum to 360 degrees.

Triangles

Triangles are polygons with three vertices at the intersection of three line segments. The interior angles of a triangle add to 180 degrees.

Isosceles triangles are triangles with two side lengths of equal measurement and two angles of equal measurement.
Equilateral triangles are those with three side lengths of equal measurement. Equilateral triangles also contain three 60-degree angles.
Right triangles contain two acute angles and a right angle. Their side lengths can be related through the Pythagorean Theorem. There are two special right triangles to become familiar with: 30−60−90 and 45−45−90 right triangles.
A 30-60-90 triangle has angle measurements of 30, 60, and 90 degrees. The side lengths of these triangles are always in the ratio of x, x√3, 2x, corresponding to the 30−60−90 angles.
A 45-45-90 triangle has angle measurements of 45, 45, and 90 degrees. The side lengths of these triangles are always in the ratio of x, x, x√2, corresponding to the 45−45−90 angles.

In order to verify that a proposed triangle is valid, compare the sum of two side lengths with the length of the third side. Two sides of a valid triangle must always be longer than the third side.

Consider the following example: Is a three-sided polygon with side lengths 4, 4, 12 a valid triangle?

Examine the sum of each two side lengths in comparison with the third length to determine whether a valid triangle can be made.

4 + 4 > 12

8 > 12

This is not true, so a triangle with the given side lengths is impossible.

Other Polygons

Regular polygons are polygons exhibiting angles and side lengths that are all equal in measure.
Squares are regular polygons because each of its angles measures 90 degrees, and all side lengths measure the same length. The perimeter of a square is equal to 4s where s is the side length. The area of a square is equal to s2.
A rectangle, another quadrilateral, is only regular when it is also a square. The perimeter of a rectangle is 2l + 2w, where l is the length and w is the width. The area of a rectangle is lw.
A parallelogram is a quadrilateral with parallel lines composing opposite sides. The properties governing transversals and parallel lines also apply to parallelograms. The perimeter of a parallelogram is found by summing the length of each side. The area of a parallelogram is equal bh, where b is the base and h is the height.
A trapezoid is a quadrilateral composed of two parallel lines. Its perimeter is found by summing each of its sides. Its area is found by multiplying the average of its bases with its height, otherwise expressed as:

A = 1/2 * (b1 + b2) * h

A rhombus is a quadrilateral that contains equal side lengths but two pairs of different angle measurements. Its perimeter is the sum of its sides and its area is its base times its height.

Congruency and Similarity of Polygons

Geometric figures sharing the same attributes (such as line segments with the same length, polygons with the same side lengths and angle measurements, etc.) are said to be congruent. Congruency is designated with this symbol, ≅.

Figures and shapes that are not the same size, but have proportional measurements are said to be similar. Similarity is designated with this symbol, ∼.

Circles

A circle is a collection of points equidistant from a point at a center.

Any line segment starting and ending on the circle that passes through the center is known as a diameter. A radius is any line segment starting at the circle’s center and ending on the circle.

Line segments that do not pass through the center of the circle are called chords.

The circumference of a circle is the perimeter, or distance around the circle. It is defined as:

C = π * d or C =2 * π * r, where d is the diameter and r is the radius.

An arc is a part of the circumference designated by two or three points.

Arc length is defined mathematically in radians through the formula: s = r * θ, where r is the radius, and θ is the measurement of the central angle. It is defined in degrees with the formula:

s = r * θ * (π/180)

A sector is the area included inside of a central angle. It is defined mathematically in radians by the formula:

A = (θ/2) * r2, where θ is the central angle measurement and r is the radius. The same area is represented in degrees through the following:

A = (θ/360) * π * r2

A central angle is any angle with its side lengths formed from two radii and an arc. An inscribed angle is any angle formed from two chords and an arc.

A tangent line is a line forming a 90-degree angle with a radius of the circle at the point along the circle’s edge.

Geometric Notation

Become familiar with the following notation:

Arc and Circle Notation

The arc of a circle is a portion of the circle’s circumferences connecting two points, it is represented with two or three points along the circle with a curved bar above them, for example: ⏜CH represents arc CH

Circles are referenced through their center and can be represented by a circle with a point inside of it.

Pythagorean Theorem

The Pythagorean Theorem relates the three sides of a right triangle.

The shorter sides of a right triangle are known as the legs, and the longest side is known as the hypotenuse. The Pythagorean Theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse:

a2 + b2 = c2

where a and b are legs, and c is the hypotenuse.

Trigonometry

The trigonometric functions sine, cosine, and tangent relate the side lengths of a right triangle to its angles.

In any right triangle, there are two legs, two acute angles, a hypotenuse, and a right angle. The trigonometric functions are defined as follows:

sin⁡θ = o/h

cos⁡θ = a/h

tan⁡θ = o/a

Where θ is one of the acute angles, o is the side opposite the angle θ, a is the side adjacent to the angle θ, and h is the hypotenuse.

Given an angle and any side length, any right triangle can be solved by using the appropriate trigonometric function.

Consider the right triangle with side lengths 3, 4, and 5. Because all side lengths are known, any angle can be found by solving any trig. function for θ. This is accomplished by evaluating the inverse trig. function: sin⁡θ = 3/5, where θ is the angle to be solved for, 3 is a side length, and 5 is the hypotenuse. In order to solve, θ must be first liberated from the function; to do so, evaluate the inverse, or arcsin of both sides:

arcsin⁡(sin⁡θ) = arcsin⁡(3/5)

θ = arcsin⁡(3/5), which can be evaluated using a scientific calculator

The same method can be used to evaluate for an unknown side length when given an angle measurement and a side length.

The expression SOHCAHTOA can be used to recall the trig. functions. SOH indicates opposite over hypotenuse, CAH indicates adjacent over hypotenuse, and TOA indicates opposite over adjacent. The first letter of each word part stands for sine, cosine, and tangent.

Other Functions

In addition to the sine, cosine, and tangent functions, it is important to develop familiarity with the secant, cosecant, and cotangent functions in addition to the inverse of each of these.

The secant function is defined as 1/cos⁡θ. Given that the cosine function is adjacent side divided by hypotenuse, the secant function is the reciprocal, or hypotenuse divided by adjacent side.

The cosecant function is defined as 1/sin⁡θ. Given that the sine function is the opposite side divided by the hypotenuse, the cosecant function is the reciprocal, or hypotenuse divided by the opposite side.

The cotangent function is defined as 1/tan⁡θ. Given that the tangent function is opposite side over adjacent side, the cotangent function is the reciprocal, or adjacent side divided by opposite side.

The inverse of each trig. function can be taken in order to liberate the angle measurement from the trig. function. For example, arcsin⁡sin⁡θ = θ.

Complementary Angle Relationship

A straight line measures 180 degrees. Consider line AB, AB↔, with ray XY, XY→, emanating from point X, which is between A and B. If angle a (∠a) and angle b (∠b) are the angles on either side of ray XY, then ∠a + ∠b = 180∘.

Angles that combine to 180 degrees are known as supplementary angles.

In a similar fashion, angles that combine to 90 degrees are known as complementary angles.

Consider the following: ∠X = 3x + 2 and ∠Y = x + 6. The two angles are complementary. What is the value of x?

Because the angles are complementary, their sum is equal to 90 degrees. Set up an equation and solve for the unknown x:

∠X + ∠Y = 90∘

3x + 2 + x + 6 = 90

4x + 8 = 90

4x = 82

x = 412

Unit Circle

To better work with and understand the relationship between circles and trigonometry, it is highly recommended that the unit circle is explored.

The unit circle is a circle with a radius of 1 that connects the coordinate plane, right triangles, trigonometric functions, degree measurement, and radian measurements. When fully described, it shows a circle divided into a collection of right triangles of various side length measurements, with a vertex lying on the edge of a circle. The coordinates of each vertex along the circle represent cosine, and sine values that satisfy the Pythagorean Formula, x2 + y2 = 1, or in trigonometric form, (cos⁡x)2 + (sin⁡y)2 = 1.

The vertices of these triangles also correspond with degree and radian measurements ranging from 0 to 2π or 360∘. You are encouraged to develop familiarity with the unit circle independently, attempting to understand its design rather than memorizing its form.

Complex Numbers

Definition and Standard Form

Complex numbers include either a real number, an imaginary number, or the combination of both, in the form of a+bi, where a is the real portion, and bi is the complex portion, with i = √−1.

Addition and Subtraction

To add or subtract complex numbers, combine the real portion of the complex numbers and the imaginary portion of the complex numbers separately. Consider the following problem:

3 − 4i + −4 + 5i

Begin by breaking the problem into the addition of the real portions:

3 + (−4) = −1

Then combine the imaginary portion:

−4i + 5i = i

Finally, combine the real and imaginary portions:

−1 + i

Multiplication

Multiplication of complex numbers entails the distribution of one complex number (both the real and imaginary portion) through the other complex number. Consider the following multiplication problem:

(3 + 2i)(−1 − 5i)

Apply the distributive property and multiply the first term through the second parenthesis:

3 * (−1) + 3 * (−5i) = −3 − 15i

Next distribute the imaginary portion through the second parenthesis:

2i * (−1) + 2i * (−5i) = −2i − 10i^2 = −2i − 10(−1) = −2i + 10

Notice that the i^2 was replaced with −1, this is because i = −1 so i2 = (−1)^2 = −1.

Finally, combining all like terms:

−3 − 15i − 2i + 10 = 7 − 17i

Conjugate

The conjugate of a complex number is another complex number that is equivalent except for the sign of the complex portion. For example, the complex conjugate of the complex number 6 + 2i is 6 − 2i.

Complex conjugates are used to rationalize complex expressions.

Division

Dividing complex numbers entails utilizing the complex conjugate in order to rationalize the denominator. Consider the complex fraction:

(2 − 2i)/(3 + 5i)

By multiplying the numerator and denominator by the conjugate of the denominator, the denominator can be converted to a real number:

(2 − 2i)/(3 + 5i) * (3 − 5i)/(3 − 5i) = (6 − 10i − 6i − 10i2)/(9 − 15i + 15i − 25i2) = (16 − 16i)/34

Page updated

Google Sites

Report abuse