# Exploring Ideas of Algebra and Coordinate Geometry

**13.4 Connecting Algebra and Geometry
13.4.1. Finding the midpoint of a line segment **

13.4.1.1. Theorem for coordinates of the midpoint : If the coordinates for the

endpoints of are for P and for Q, then the

coordinates of the midpoint , M, of are

**13.4.2. Finding the distance between two points**

13.4.2.1.

**Pythagorean Theorem**= a

^{2}+ b

^{2}= c

^{2}

13.4.2.1.1. for right triangles only

13.4.2.1.2. a and b are legs of the triangle – the sides that form the right

angle

13.4.2.1.3. c is the hypotenuse – the side opposite the right angle and

the longest side of the triangle

13.4.2.2.

**Distance Formula**based on the Pythagorean Theorem

13.4.2.3.

**Distance Formula:**If d is the distance between points and

, then

**13.4.3. Finding the equation of a line**

13.4.3.1. Slope- intercept method

13.4.3.1. Slope- intercept method

13.4.3.1.1. Slope is designated as “m”

13.4.3.1.1.1.

13.4.3.1.2. Slope- Intercept form of a line

13.4.3.1.2.1. (0,b) is the y-intercept

13.4.3.1.2.2. y = mx + b

**13.4.3.2. Two point method**

13.4.3.3. Given and , then

13.4.3.3.1. First, solve for m to obtain the slope of the line

13.4.3.3.2. Next choose (arbitrarily could choose if

desired, but can ONLY choose one point)

13.4.3.3.3. Then and solve for y to put the equation

into slope-intercept form

13.4.3.3.4. Leave fractions in improper form – do

**NOT**convert to mixed

numbers

**13.4.3.4. Point slope method**

13.4.3.4.1. Given m and , the equation for a line can be found

by:

13.4.3.4.2. Solve for y to put the equation into slope-intercept form

**13.4.3.5. Finding the point of intersection of two lines**

13.4.3.5.1. First find the equation of each line: and

13.4.3.5.2. Next, use substitution to solve for x:

13.4.3.5.3. After finding x, choose either original equation to solve for y,

say

13.4.3.5.4. The x and y you found are the (x ,y) of the point of

intersection between the two lines

**13.4.3.5.5. What happens when the slopes are equal?**

13.4.3.6. Horizontal and vertical lines

13.4.3.6.1. Horizontal line properties

13.4.3.6. Horizontal and vertical lines

13.4.3.6.1. Horizontal line properties

13.4.3.6.1.1. The slope of every horizontal line is in the form: y = b,

where (0,b) is the y-intercept

13.4.3.6.1.2. The slope of every horizontal line is zero

**13.4.3.6.2. Vertical line properties**

13.4.3.6.2.1. The slope of every vertical line is in the form: x = t,

where (t,0) is the x-intercept

13.4.3.6.2.2. The slope of every vertical line is undefined; a vertical

line has no slope

**13.4.4. Using coordinate geometry to verify geometric conjectures**

13.4.4.1. skip

**13.4.5. Developing the equation of a circle**

13.4.5.1. Circle equation theorem 1: The standard form for an equation of a

circle with a center at the origin and radius r is: x

^{2}+ y

^{2}= r

^{2}

13.4.5.2. Circle equation theorem 2: The standard form for an equation of a

circle with a center at (h, k) and radius r is:

**13.4.6. Describing transformations using coordinate geometry**

13.4.6.1. skip for now – will include with transformations later

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