1 2 Measuring and Constructing Segments Objectives Use

1 -2 Measuring and Constructing Segments Objectives Use length and midpoint of a segment. Construct midpoints and congruent segments. Holt Geometry

1 -2 Measuring and Constructing Segments Holt Geometry

1 -2 Measuring and Constructing Segments The distance between any two points is the absolute value of the difference of the coordinates. If the coordinates of points A and B are a and b, then the distance between A and B is |a – b| or |b – a|. The distance between A and B is also called the length of AB, or AB. A a Holt Geometry B b AB = |a – b| or |b - a|

1 -2 Measuring and Constructing Segments Example 1: Finding the Length of a Segment Find each length. A. BC B. AC BC = |1 – 3| Holt Geometry AC = |– 2 – 3| = |1 – 3| = |– 5| =2 =5

1 -2 Measuring and Constructing Segments Check It Out! Example 1 Find each length. a. XY Holt Geometry b. XZ

1 -2 Measuring and Constructing Segments Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write PQ RS. This is read as “segment PQ is congruent to segment RS. ” Tick marks are used in a figure to show congruent segments. Holt Geometry

1 -2 Measuring and Constructing Segments You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge. Holt Geometry

1 -2 Measuring and Constructing Segments Example 2: Copying a Segment Sketch, draw, and construct a segment congruent to MN. Step 1 Estimate and sketch. Estimate the length of MN and sketch PQ approximately the same length. Holt Geometry P Q

1 -2 Measuring and Constructing Segments Example 2 Continued Sketch, draw, and construct a segment congruent to MN. Step 2 Measure and draw. Use a ruler to measure MN. MN appears to be 3. 5 in. Use a ruler to draw XY to have length 3. 5 in. Holt Geometry X Y

1 -2 Measuring and Constructing Segments Example 2 Continued Sketch, draw, and construct a segment congruent to MN. Step 3 Construct and compare. Use a compass and straightedge to construct ST congruent to MN. A ruler shows that PQ and XY are approximately the same length as MN, but ST is precisely the same length. Holt Geometry

1 -2 Measuring and Constructing Segments In order for you to say that a point B is between two points A and C, all three points must lie on the same line, and AB + BC = AC. Holt Geometry

1 -2 Measuring and Constructing Segments Example 3 B: Using the Segment Addition Postulate M is between N and O. Find NO. NM + MO = NO 17 + (3 x – 5) = 5 x + 2 3 x + 12 = 5 x + 2 – 2 3 x + 10 = 5 x – 3 x 10 = 2 x 2 2 5=x Holt Geometry Seg. Add. Postulate Substitute the given values Simplify. Subtract 2 from both sides. Simplify. Subtract 3 x from both sides. Divide both sides by 2.

1 -2 Measuring and Constructing Segments Example 3 B Continued M is between N and O. Find NO. NO = 5 x + 2 Holt Geometry = 5(5) + 2 Substitute 5 for x. = 27 Simplify.

1 -2 Measuring and Constructing Segments The midpoint M of AB is the point that bisects, or divides, the segment into two congruent segments. If M is the midpoint of AB, then AM = MB. So if AB = 6, then AM = 3 and MB = 3. Holt Geometry

1 -2 Measuring and Constructing Segments Example 5: Using Midpoints to Find Lengths D is the midpoint of EF, ED = 4 x + 6, and DF = 7 x – 9. Find ED, DF, and EF. E 4 x + 6 Step 1 Solve for ED = DF 4 x + 6 = 7 x – 9 – 4 x D 7 x – 9 F x. D is the mdpt. of EF. Substitute 4 x + 6 for ED and 7 x – 9 for DF. Subtract 4 x from both sides. 6 = 3 x – 9 Simplify. +9 + 9 Add 9 to both sides. Simplify. 15 = 3 x Holt Geometry

1 -2 Measuring and Constructing Segments Example 5 Continued D is the midpoint of EF, ED = 4 x + 6, and DF = 7 x – 9. Find ED, DF, and EF. E 4 x + 6 15 3 x = 3 3 x=5 Holt Geometry D 7 x – 9 Divide both sides by 3. Simplify. F

1 -2 Measuring and Constructing Segments Example 5 Continued D is the midpoint of EF, ED = 4 x + 6, and DF = 7 x – 9. Find ED, DF, and EF. E 4 x + 6 D 7 x – 9 F Step 2 Find ED, DF, and EF. ED = 4 x + 6 DF = 7 x – 9 = 4(5) + 6 = 7(5) – 9 = 26 Holt Geometry EF = ED + DF = 26 + 26 = 52