Chapter 2. The Midpoint Formula:

Transcription

1 Chapter 2 The Midpoint Formula: Sometimes you need to find the point that is exactly between two other points. For instance, you might need to find a line that bisects (divides into equal halves) a given line segment. This middle point is called the "midpoint". The concept doesn't come up often, but the Formula is quite simple and obvious, so you should easily be able to remember it for later. Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is [5 + 10] / 2 = 15 / 2 = 7.5. The Midpoint Formula works exactly the same way. If you need to find the point that is exactly halfway between two given points, just average the x-values and the y-values. Ex: Find the midpoint between ( 1, 2) and (3, 6). 1 P a g e

2 Technically, the Midpoint Formula is the following: But as long as you remember that you're averaging the two points' x- and y-values, you'll do fine. It won't matter which point you pick to be the "first" point you plug in. Ex: Find the midpoint between (6.4, 3) and ( 10.7, 4). Ex: Find the value of p so that ( 2, 2.5) is the midpoint between (p, 2) and ( 1, 3). 2 P a g e

3 Ex: Find the center of the circle with a diameter having endpoints at ( 4, 3) and (0, 2). Section An Introduction to Applications of Linear Equations 3 P a g e

4 Objectives: 1. Solve problems involving supplementary and complementary angles. Solving an Applied Problem Step 1 Read the problem, several times if necessary, until you understand what is given and what is to be found. Step 2 Assign a variable to represent the unknown value, using diagrams or tables as needed. Write down what the variable represents. Express any other unknown values in terms of the variable. Step 3 Write an equation using the variable expression(s). Step 4 Solve the equation. Step 5 State your answer. Does it seem reasonable? Step 6 Check the answer in the words of the original problem. Solving with Supplementary and Complementary Angles 4 P a g e

5 Problem-Solving Hint If x represents the degree measure of an angle, then 90 x represents the degree measure of its complement, and 180 x represents the degree measure of its supplement. Ex: Find the measure of an angle whose supplement is 20 more than three times its complement. 5 P a g e

6 Ex: The supplement of an angle measures 10 times more than the measure of its complement. What is the measure of the angle (in degrees)? Ex: Find the measure of an angle whose supplement is 10 degrees more than twice its complement. 6 P a g e

7 Section Formulas and Additional Applications from Geometry Objectives: 1. Solve a formula for one variable, given the values of the other variables. 2. Use a formula to solve an applied problem. Ex: Find the value of the remaining variable. P = 2L + 2W; P = 52; L = 8 Ex: Find the value of the remaining variable for the following ( ) 7 P a g e

8 Ex: The area of a rectangular garden is 187 in 2 with a width of 17 in. What is the length of the garden? 8 P a g e

9 Ex: Bob is working on a sketch for a new underwater vehicle (UV), shown below. In his sketch, the bottom of the UV is 10 ft long, the top is 8 ft long, and the area is 63 ft 2. What is the height of his UV? Ex: The area of a triangular sail is 126ft 2. (Recall ft 2 means square feet ) The base of the sail is 12ft. Find the height of the sail. 9 P a g e

10 Ex: The longest side of a triangle is 3ft longer than the shortest side. The medium side is 1ft longer than the shortest side. If the perimeter of the triangle is 16ft what are the lengths of the three sides. Ex: Kari Heen s backyard is the shape of a rectangle. The length is 5m less than twice the width. And the perimeter is 80m. Find the dimensions of the yard. 10 P a g e