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Unit 1 Chapter 1 1-3 Distance and Midpoints

The distance between two points is the length of the segment with those points as its endpoints. Concept

Use the number line to find QR. Find Distance on a Number Line Use the number line to find QR. The coordinates of Q and R are –6 and –3. QR = | –6 – (–3) | Distance Formula = | –3 | or 3 Simplify. Answer: 3 Example 1

To find the distance between two points A and B in the coordinate plane, you can form a right triangle with as its hypotenuse and point C as its vertex as shown. The use the Pythagorean Theorem to find AB.

Since the formula for finding the distance between two points involves taking the square root of a real number, distances can be irrational. An irrational number is a number that cannot be expressed as a terminating or repeating decimal. Concept

Find the distance between E(–4, 1) and F(3, –1). Find Distance on a Coordinate Plane Find the distance between E(–4, 1) and F(3, –1). (x1, y1) = (–4, 1) and (x2, y2) = (3, –1) Example 2

Find Distance on a Coordinate Plane Check Graph the ordered pairs and check by using the Pythagorean Theorem. Example 2

Find Distance on a Coordinate Plane . Example 2

The midpoint of a segment is the point halfway between the endpoints of the segment. If X is the midpoint of , then AX = XB and . You can find the midpoint of a segment on a number line by finding the mean, or the average of the coordinates of its endpoints. Concept

Find Midpoint on a Number Line DECORATING Marco places a couch so that its end is perpendicular and 2.5 feet away from the wall. The couch is 90” wide. How far is the midpoint of the couch back from the wall in feet? First we must convert 90 inches to 7.5 feet. The coordinates of the endpoints of the couch are 2.5 and 10. Let M be the midpoint of the couch. Midpoint Formula x1 = 2.5, x2 = 10 Example 3

Answer: The midpoint of the couch back is 6.25 feet from the wall. Find Midpoint on a Number Line Simplify. Answer: The midpoint of the couch back is 6.25 feet from the wall. Example 3

Concept

Find Midpoint in Coordinate Plane Answer: (–3, 3) Example 4

Let D be (x1, y1) and F be (x2, y2) in the Midpoint Formula. Find the Coordinates of an Endpoint Let D be (x1, y1) and F be (x2, y2) in the Midpoint Formula. (x2, y2) = (–5, –3) Write two equations to find the coordinates of D. Example 5

Answer: The coordinates of D are (–7, 11). Find the Coordinates of an Endpoint Midpoint Formula Midpoint Formula Answer: The coordinates of D are (–7, 11). Example 5

Use Algebra to Find Measures Understand You know that Q is the midpoint of PR, and the figure gives algebraic measures for QR and PR. You are asked to find the measure of PR. Example 6

Plan Because Q is the midpoint, you know that Use Algebra to Find Measures Plan Because Q is the midpoint, you know that Use this equation and the algebraic measures to find a value for x. Solve Subtract 1 from each side. Example 6

Use Algebra to Find Measures Original measure Example 6

QR = 6 – 3x Original Measure Use Algebra to Find Measures Check QR = 6 – 3x Original Measure Example 6

Use Algebra to Find Measures Multiply. Simplify. Example 6

Any segment, line, or plane that intersects a segment at its midpoint is called a segment bisector.   In the figure at the right, M is the midpoint of . Plane A, ,and point M are all bisectors of .