1. Solve 3x = 8x – 15. ANSWER 3 2. Solve 6x + 3 = 8x – 14. ANSWER 8.5 3. If M is the midpoint of AB, AM = 5x – 2, and MB = 3x + 6, find AB. ANSWER 36

Use properties of special segments in triangles. Target Use properties of special segments in triangles. You will… Use perpendicular bisectors to solve problems.

Vocabulary perpendicular bisector – a segment, ray, line, or plane that is perpendicular to a given segment at its midpoint. equidistant – a point that is the same distance from two figures P is equidistant from A and B and lies on the perpendicular bisector of AB … as does C (see below) Perpendicular Bisector Theorem 5.2 – In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Perpendicular Bisector Converse Theorem 5.3 – In a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector.

Vocabulary concurrent lines – three or more lines that intersect in the same point (point of concurrency); also applies to rays and segments. Concurrency of Perpendicular Bisectors of a Triangle Theorem 5.4 – The perpendicular bisectors of a triangle intersect at a point (circumcenter) that is equidistant from the vertices of the triangle. P is the circumcenter of triangle ABC

Use the Perpendicular Bisector Theorem EXAMPLE 1 Use the Perpendicular Bisector Theorem ALGEBRA BD is the perpendicular bisector of AC . Find AD. AD = CD Perpendicular Bisector Theorem 3x + 14 5x = Substitute. 7 x = Solve for x. AD = 5x = 5(7) = 35.

EXAMPLE 2 Use perpendicular bisectors In the diagram, WX is the perpendicular bisector of YZ . a. What segment lengths in the diagram are equal? b. Is V on WX ? SOLUTION a. WX bisects YZ , so XY = XZ. Because W is on the perpendicular bisector of YZ, WY = WZ by Perpendicular Bisector Theorem 5.2. The diagram shows that VY = VZ = 25. b. Because VY = VZ, V is equidistant from Y and Z. So, by the Converse of the Perpendicular Bisector Theorem, V is on the perpendicular bisector of YZ , which is WX .

GUIDED PRACTICE for Examples 1 and 2 In the diagram, JK is the perpendicular bisector of NL . 1. What segment lengths are equal? Explain your reasoning. 2. Find NK. NJ =LJ since JK bisects NL. NK = LK by the Perpendicular Bisector Theorem and the diagram shows ML = MN. ANSWER 13 ANSWER

GUIDED PRACTICE for Examples 1 and 2 In the diagram, JK is the perpendicular bisector of NL . 3. Explain why M is on JK . Since ML = MN, M is equidistant from N and L, so by the Converse of the Perpendicular Bisector Theorem M is on the perpendicular bisector of NL which is JK. ANSWER

EXAMPLE 3 Use the concurrency of perpendicular bisectors Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor. FROZEN YOGURT Find a location for the distributor that is equidistant from the three carts.

EXAMPLE 3 Use the concurrency of perpendicular bisectors Theorem 5.4 shows you that you can find a point equidistant from three points by using the perpendicular bisectors of the triangle formed by those points. Copy the positions of points A, B, and C and connect those points to draw ABC. Then use a ruler and protractor to draw the three perpendicular bisectors of ABC. The point of concurrency D is the location of the distributor.

GUIDED PRACTICE for Example 3 4. WHAT IF? Hot pretzels are sold from points A and B and also from a cart at point E. Where could the pretzel distributor be located if it is equidistant from those three points? Draw the triangle and show the location. Where the perpendicular bisectors of the triangle formed by A, B, and C intersect ANSWER