Unit 7 Advanced Geometry Project - Tristan and Billy
This book is a compilation of topics from Unit 7: Coordinate Geometry. The topics include: Distance, Midpoint, Slope, Slope-intercept form, Standard form, Area, and Perimeter.
Table of Contents
- Distance and its Formula
- Distance and its Forumla
- Midpoints
- Midpoints
- Slope
- Slope
- Slope-Intercept Form
- Slope-Intercept Form
- Standard Form
- Standard Form
- Area
- Area
- Perimeter
- Perimeter
Distance and its Forumla
Distance and its Formula
Finding the distance between two points isn't as hard as the formula may seem. The distance formula is: d=√(x[sub]2[/sub]−x[sub]1[/sub])[sup]2[/sup]+(y[sub]2[/sub]−y[sub]1[/sub])[sup]2[/sup]. When solving for the distance between two points, you can also make a triangle (on a coordinate plane) and find the rise and run of the legs. Then, you can use Pythagorean Theorem (a[sup]2[/sup]+b[sup]2[/sup]=c[sup]2[/sup])to find the hypotenuse, the straight-line distance between the two points.
Try it for Yourself #1 - Find the question mark!
In the problem above, you are trying to find the hypotenuse, or the distance between points A and B. So, you use the Pythagorean Theorem and solve for the missing side. Since you know the two legs, square each of them (BC=16, AC=25), add them, and then take the square root of them added together. Thus, the distance between A and B is the square root of 41, which equals roughly 6.4.
Midpoints
Midpoints
A midpoint is exactly what it sounds like: a midpoint or a central point on a line. The basic midpoint formula is: (x[sub]2[/sub]+x[sub]1[/sub])/2, (y[sub]2[/sub]+y[sub]1[/sub])/2. You can also be given an endpoint and the MIDPOINT, not the other endpoint. If this happens, you take similar steps. You take x value of the endpoint that you do know and add "x". Then, you divide by 2 and set it all equal to the x value of the midpoint. Next, you go on to do the same thing for the y coordinate, but you switch the y values according to the endpoint and midpoint.
Try it yourself: Find the Midpoint!
So first, find the x coordinates of the two endpoints, in this case, 2 and -1. Take x[sub]2[/sub] and add the x[sub]1[/sub](2+(-1)). Then, divide this by two, and for the x value of the midpoint, you get 0.5! Next, do the same thing for the y coordinates: y[sub]2[/sub]+y[sub]1[/sub] (5+2). Once again, divide this by two, and the y value of the midpoint is 3.5! Finally, put them together, and the midpoint equals (0.5,3.5).
Slope
Slope
Slope, is essentially the steepness of a line, expressed at a number, with higher numbers meaning steeper lines and bigger slopes, and lower numbers representing lines that go up gradually and lower slopes. There are also both positive and negative slopes, with positive slopes rising from left to right and negative slopes falling from left to right. To find the slope of a line (when given coordinates of two points on that line), you use the Slope Formula. The formula is (y[sub]2[/sub] - y[sub]1[/sub]) / (x[sub]2[/sub] - x[sub]1[/sub]). Try this example:
First, lets solve the y[sub]2[/sub]-y[sub]1[/sub] part of this. The y[sub]2[/sub] value is 5, and the y[sub]1[/sub] value is 4, so you subtract 4 from 5, which equals 1. Next, lets solve for the x[sub]2[/sub]-x[sub]1[/sub]. [size=85][size=100]The x[sub]2[/sub] value is 2, and the x[sub]1[/sub] value is -2, so you subtract -2 from 2, which eqauls 4. So, all in all, the slope of the line is 1/4 or 0.25.[/size][/size]
Slope-Intercept Form
Slope-Intercept form tells you the slope and the y-intercept of a line. The slope-intercept equation is y=mx+b, where "m" stands for the slope and "b" is the y-intercept. The only way a line cannot be represented by slope-intercept form is if it is a vertical or horizontal line. For example vertical lines can be represented by x=5, while horizontal lines are represented by y=4.
Find the slope and y-intercept!
First you should find the y-intercept since this is the easiest part of the equation, look vertically over the y axis and you should see where the line meets the y axis. So, the y-intercept of the equation is 5, because it is where it intersects the y-axis. Next you will find the slope, which is rise over run, meaning y will be on top divided by x. The slope is 2 because it goes up 2 and over 1 before it hits the line again!
Standard Form
Standard Form
Standard Form is another way to write the equation of a line. The Standard Form equation is Ax+By=C. A, B, and C all represent coefficients. It's similar to Slope-Intercept Form, however, x and y are on the same side, unlike the prior. The slope in Standard Form is represented by the "A" and the "C" represents the y-intercept. To turn Standard Form into Slope-Intercept Form, you add or subtract (depending on whether A is positive or negative) the "Ax" to the other side, so you get By=-Ax+C. Then, you divide "[math]\pm[/math]Ax+C" by B, and you get your equation into Slope-Intercept Form. To turn Slope-Intercept into Standard Form, you add or subtract (depending if m is positive or negative) the "mx" to the other side so that you get "[math]\pm[/math]Ax+By=C." Another helpful tip to remember when learning Standard Form is that "A" should NOT be negative. Try this practice problem below!
Try finding this in Slope-Intercept Form, and transfer it into Standard Form!
First, find the Slope-Intercept Form equation of the line. Remember, to find the slope, do y[sub]2[/sub]-y[sub]1[/sub]/x[sub]2[/sub]-x[sub]1[/sub]. When you do this, you should get 2-5/-1-2, which equals a slope of -3/-3, or 1. Then, you need to find the y-intercept. This is the point where the line hits the y-axis. On this problem, that point is at (0,3). Now, you have a Slope-Intercept equation of y=1x+3. Then, follow the steps that I talked about above to transfer this equation into Standard Form. This is pretty easy to remember, because you just need to subtract 1x from each side. You know have an equation of -1x+y=3. Also, since A cannot be negative, you need to divide the whole equation then by -1 (just for this case) so that your A value can be positive. After doing this, you have your Standard Form equation of 1x-y=-3.
Area
Area
The area of a figure tells how much space the shape takes up. You probably already know the area of a square (side times side) and a rectangle (length times width). Here are the equations to find some other common shapes' areas... 1) Triangle: 1/2 base times height (1/2(bh)), 2) Trapezoid: 1/2 height (base 1 plus base 2) (1/2 times (height times (base 1 plus base 2))), and 3) Parallelogram: base times height. When doing area problems for triangles, you need to make sure to rid of the extra space around the triangle (after drawing a box around it such as in the problem below). To do this, multiply how high and wide the box needs to go to accommodate all points of the triangle. For example, in the problem below, the area before subtracting the extra space is 56.
Triangle Area Problem
Triangle Area Solved
Next, multiply each of the three parts of the triangle that have blank space together. So, you would multiply 8 by 3, 7 by 5, and 3 by 4. Multiply each of those answers by 1/2... and subtract all three from 56! When you do this, you should get 56-(12+17.5+6), which equals an area of 20.5 units [sup]2[/sup].
Parallelogram Problem
Parallelogram Problem Solved
This one is pretty easy. Remember, the formula to find the area of a parallelogram is simply base times height. In the case of a parallelogram, you don't need to subtract the empty space because the height isn't changing, meaning that you can go anywhere horizontally in the figure and the height is still 7 units. So, all you need to do to find the area of this figure is multiply 7 by 7, and your area is 49 units[sup]2[/sup].
Perimeter
Perimeter is the length around a figure. The way to find perimeter is the distance of each side added together (for squares). But in the case of a triangle you would find the length of each side of the triangle/parallelogram (for lines you can just the length of the line, and for non-straight lines such as the vertical lines of a parallelogram where the point that the line is on the x-axis changes, and do the use the distance formula to find those uneven lines). Then add each up, the sides can be found by counting how farm along the y and x axis it traveled then set it up as this equation, x = [math]\sqrt{Rise^2+Run^2}[/math].
Try to solve for the perimeter of a scalene triangle!
First to solve for the perimeter you will have to find the length of each side. For example side has a rise of -1 and a run of 4, so the equation would be [math]\sqrt{-1^2+4^2}=4.12[/math], where 4.12 is the length of a. Then you would do the same for the other sides, b and c, which b should equal 5 and c should equal 3.16. Then you would add them all together so 3.16+5+4.12=11.28.