1.3: Use Midpoint and Distance Formulas

1.3: Use Midpoint and Distance Formulas Objectives: • To define midpoint and segment bisector • To use the Midpoint and Distance Formulas • To construct a segment bisector with a compass and straightedge

Perpendicular Bisector • Draw a segment. Label the endpoints A and B.

Perpendicular Bisector • Using the same compass setting, draw two intersecting arcs through the segment, one centered at A, the other centered at B. Label the intersection points C and D.

Perpendicular Bisector • Draw a line through points C and D.

Perpendicular Bisector • Label the new point of intersection M. Is point is called the midpoint.

Perpendicular Bisector: Video Click on the button to watch a video of the construction.

Vocabulary In you notes, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

Midpoint The midpoint of a segment is the point on the segment that divides, or bisects, it into two congruent segments.

Segment Bisector A segment bisector is a point, ray, line, line segment, or plane that intersects the segment at its midpoint.

Example 1 Find DM if M is the midpoint of segment DA, DM = 4x – 1, and MA = 3x + 3. Work it out, labeling the parts of the drawing 4x-1 = 3x+3 X = 4 4(4)-1 15

Example 2: SAT In the figure shown, ABCD is a rectangle with BC = 4 and QR = 6. Points P, Q, and R are different points on a line (not shown) that is parallel to AD. Points P and Q are symmetric about line AB and points Q and R are symmetric about line CD. What is PR? 8

Example 3 Segment OP lies on a real number line with point O at –9 and point P at 3. Where is the midpoint of the segment? What if the endpoints of segment OP were at x1 and x2? -3 1 6 4 2 4 1 6 3 3 2 5 5 -9+3 = -3 2 OR x1 + x2= midpoint (the average) 2

In the Coordinate Plane We could extend the previous exercise by putting the segment in the coordinate plane. Now we have two dimensions and two sets of coordinates. Each of these would have to be averaged to find the coordinates of the midpoint.

The Midpoint Formula If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the midpoint M of AB has coordinates

The Midpoint Formula The coordinates of the midpoint of a segment are basically the averages of the x- and y-coordinates of the endpoints

Example 4 Find the midpoint of the segment with endpoints at (-1, 5) and (3, 8). (1, 7.5)

Example 5 The midpoint C of IN has coordinates (4, -3). Find the coordinates of point I if point N is at (10, 2). So, in other words, (4,3) is the AVERAGE of (10,2) and some other point. THINK ABOUT IT! You could also graph it (-2, -8)

Example 6 Use the Midpoint Formula multiple times to find the coordinates of the points that divide AB into four congruent segments.

Parts of a Right Triangle Which segment is the longest in any right triangle? The Hypotenuse

The Pythagorean Theorem In a right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then c2 = a2 + b2.

Example 7 How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall? 52 + h2 = 202 25 + h2 = 400 h2 = √375 h = 19.4

The Distance Formula Sometimes instead of finding a segment’s midpoint, you want to find it’s length. Notice how every non-vertical or non-horizontal segment in the coordinate plane can be turned into the hypotenuse of a right triangle.

Example 8 Graph AB with A(2, 1) and B(7, 8). Add segments to your drawing to create right triangle ABC. Now use the Pythagorean Theorem to find AB. 52 + 72 = c2 25 + 49 = c2 74 = c2 8.6 = c

Distance Formula In the previous problem, you found the length of a segment by connecting it to a right triangle on graph paper and then applying the Pythagorean Theorem. But what if the points are too far apart to be conveniently graphed on a piece of ordinary graph paper? For example, what is the distance between the points (15, 37) and (42, 73)? What we need is a formula!

The Distance Formula To find the distance between points A and B shown at the right, you can simply count the squares on the side AC and the squares on side BC, then use the Pythagorean Theorem to find AB. But if the distances are too great to count conveniently, there is a simple way to find the lengths. Just use the Ruler Postulate.

The Distance Formula You can find the horizontal distance subtracting the x-coordinates of points A and B: AC = |7 – 2| = 5. Similarly, to find the vertical distance BC, subtract the y-coordinates of points A and B: BC = |8 – 1| = 7. Now you can use the Pythagorean Theorem to find AB.

Example 9 Generalize this result and come up with a formula for the distance between any two points (x1, y1) and (x2, y2).

The Distance Formula If the coordinates of points A and B are (x1, y1) and (x2, y2), then MEMORIZE this formula!

Example 10 To the nearest tenth of a unit, what is the approximate length of RS, with endpoints R(3, 1) and S(-1, -5)? WORK IT OUT! 8.9 Did you get it?

Example 11 A coordinate grid is placed over a map. City A is located at (-3, 2) and City B is located at (4, 8). If City C is at the midpoint between City A and City B, what is the approximate distance in coordinate units from City A to City C? THINK about it and work it out 4.6

Example 12 Points on a 3-Dimensional coordinate grid can be located with coordinates of the form (x, y, z). Finding the midpoint of a segment or the length of a segment in 3-D is analogous to finding them in 2-D, you just have 3 coordinates with which to work.

Example 12 Find the midpoint and the length of the segment with endpoints (2, 5, 8) and (-3, 1, 2). (-.5, 3, 5)

1.3: Use Midpoint and Distance Formulas