1) If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
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1 5.1 and 5.2 isectors in s l Theorems about perpendicular bisectors 1) If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Given: l is bisector of A P in on l Prove: PA = P A P 2) If a point is equidistant from the endpoints of a segment then the point lies on the perpendicular bisector of the segment. Given: PA = P Prove: P is on the bisector of A P A 3) The point of concurrency for the perpendicular bisectors of a triangle is the circumcenter. 4) Circumcenter Theorem: The circumcenter of a triangle is equidistant from the vertices of the triangle. 5) The circumcenter of a triangle is also the center of the circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. Theorems about angle bisectors 1) If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Given: D bisects AC DE A DF C Prove: DE = DF 2) If a point is equidistant from the sides of an angle of a triangle, then the point lies on the bisector of the angle. Given: DE = DF DE A DF C Prove: D bisects AC 3) The point of concurrency for the angle bisectors of a triangle is the incenter. 4) Incenter Theorem: The incenter of a triangle is equidistant from the sides of the triangle. 5) The incenter is the center of the triangle s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.
2 Practice Problems 1) Q is equidistant from the sides of Find the value of x. Use the figure on the right for Exercises Given that line p is the perpendicular bisector of XZ and XY 15.5, find ZY. 3. Given that XZ 38, YX 27, and YZ 27, find ZW. 4.Given that XY ZY, WX 6x 1, and XZ 10x 16, find ZW. Use the figure for Exercises Given that FG HG and m FEH 58, find m GEH. 6. Given that EG bisects FEH and GF 2, find GH. 7. Given that FEG GEH, FG 10z 30, and HG 7z 6, find FG. 8. Given that GF GH, m GEF 8 3 a, and m GEH 24, find a. Finding the circumcenter using algebra 1. Write an equation in standard form of the perpendicular bisector of AC in ΔAC A (2, 1), (-1, 3), C (6, 3). 1) Find the coordinates of the midpoint of AC and call it M. 2) Find the slope of AC. 3) Find the slope of a line perpendicular to AC. 4) Write an equation of the line through M having the slope found in (3).
3 Find the circumcenter of AC. 2. A music company has stores at A(0,0) (8,0) and C(4,3) where each unit of the coordinate plane represents one mile. a) A new store will be built so that it is equidistant from the three existing stores. Find the coordinates of the new store s location. b) Where will the new store be located in relation to AC? c) To the nearest tenth of a mile, how far will the new store be from each of the existing stores? 3. Find the circumcenter of OWL. O (0,0) W(0,19) L(-3,0) 4. Find the circumcenter of FIG. F (0,0) I (0,12) G (6,6) 5.3 Medians and Altitudes in s A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side. The point of concurrency of the medians of a triangle is the centroid. Centroid Theorem The medians of a are concurrent at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side. The altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. The point of concurrency of the altitudes of a triangle is the orthocenter.
4 Practice Problems: 1) Example 2) name the median, altitude, and angle bisector Example 3) Find the value of the variables SU is a median In PRS, PT is an altitude and PX is a median. 1. Find RS if RX = x + 7 and SX = 3x Find RT if RT = x 6 and m PTR = 8x Find x if EG is a median of DEF.
5 Finding the centroid and orthocenter using algebra 1. Write an equation in standard form of the median from in ΔAC, given the coordinates as follows: A (2, 1), (-1, 3), C (6, 3). 1) Find the coordinates of the midpoint of AC and call it M 2) Find the slope of M. 3) Write the equation of the line from to M. 2. Write an equation in standard form of the altitude from in ΔAC in part I. 1) Find the slope of AC 2) Find the slope of the altitude, i.e. find the slope of a line perpendicular to AC. 3) Write an equation of the line through having the slope found in (2). 3. Given ΔAC with A (4, 4), (6, 2), and C (-2, -4), write an equation of the median from A. 4. Given ΔAC with A (-1, 5), (-7, -3), and C (5, 1), write an equation of the altitude from A to C. 5. Given ΔAC with A (0, 0), (2, 4), and C (-4, 2), write an equation of the altitude to C. 6. The vertices of ΔDEF are D(5,5), E(5,-4), F(-1,-1). Find the coordinates of the Orthocenter. 7. The vertices of ΔDEF are D(-1,5), E(7,2), F(-1,-4). Find the coordinates of the Orthocenter. 8. The vertices of ΔMNO are M(-2,5), N(6,-3), O(2,-5). Find the coordinates of the Centroid.
6 5.4 The mid-segment of a triangle is a segment joining the midpoints of two sides of a triangle. Properties of a mid segment: 1. is to the third side 2. is as long as the third side. Practice Problems M, N, and P are midpoints of XZ,ZY, and XY, respectively. 1.) Mark the diagram with tick marks: 2) Name all s: 3) XY // ; XZ // ; MP // X P Example 1) Given DE, DF, and FE are the lengths of Example 2) Given AC = 42, C = 46, mid-segments. Find the perimeter of triangle AC. A = 48, D, E, and F are midpoints Find the perimeter of triangle DEF M Z N Y Example 3) D and E are midpoints. Find m<a and Example 4) Points, D, and F are midpoints. m< EDA. EC = 30 and DF = 23. Find AC. Example 5) Find the value of x. Example 6) Identify the mid-segment and find its length. Example 7) If E = 2x+6 and DF = 5x+9, find the value of x, DF, and E 6
7 5.5 and 5.6 Inequalities (1 and 2 Triangles) Theorem: In a, the smallest is opposite the shortest side. Theorem: In a, the largest is opposite the longest side. 54 D 13mm 12mm E Converses are also true! 52 11mm Theorem: In a, the shortest side is opposite the smallest. Theorem: In a, the longest side is opposite the largest. C F 1. In AC name the sides in order from least to greatest. 2. In DEF name the angles in order from greatest to least. 3. Name the shortest and longest sides in right FIT if F is the right angle and m I = 48. Theorem The sum of the lengths of any 2 sides of a triangle is greater than the length of the 3 rd side. 4. Can a triangle have sides with the given lengths? a) 4m, 7m, 8m d) 1.2cm, 2.6cm, 4.9cm b) 4in, 4in, 4in e) 11m, 12m, 14m c) 18ft, 20ft, 40ft f) 2.5m, 3.5m, 6m The lengths of two sides of a triangle are given. Find the range of possible lengths of the third side m, 3.5 m ft, 177 ft mi, 4 mi 7
8 Practice Problems In the following exercises the diagrams are not drawn to scale. If each diagram were drawn to scale, which numbered angle would be largest? Which segment would be x+ 1 the largest? x x - 1 8
9 Compare the given measures. 1. m K and m M 2. A and DE 3. QR and ST Find the range of values for x The diagrams are not drawn to scale. Which numbered angle would be the largest? cm 6 cm cm y 1 Which segment is the longest? C 63 3 y - 2 y a c b A 58 A (x + 1) (x + 3) C 9
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