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 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 P in on l Prove: P = P l l P 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: P = P Prove: P is on the bisector of l P P 3) The point of concurrency for the perpendicular bisectors of the sides 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. 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 C DE 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 DF C Prove: D bisects C 3) The point of concurrency for the angle bisectors of a triangle is the incenter. 1
2 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. circle inscribed in a polygon is tangent to (intersects at one point) each side of the triangle. 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 a, and m GEH 24, find a. 3 2
3 Finding the circumcenter using algebra 1. Write an equation in standard form of the perpendicular bisector of C in ΔC (2, 1), (-1, 3), C (6, 3). 1) Find the coordinates of the midpoint of C and call it M. 2) Find the slope of C. 3) Find the slope of a line perpendicular to C. 4) Write an equation of the line through M having the slope found in (1). 3. Find the equation of the perpendicular bisector of side WL in OWL. O (0,0) W(0,19) L(-3,0) 4. Find the equation of the perpendicular bisector of side IG in FIG. F (0,0) I (0,12) G (6,6) 5.3 Medians and ltitudes in s 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. n altitude of a triangle is a 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. Practice Problems: 1) Example 2) name the median, altitude, and angle bisector 3
4 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. 4
5 Equations of Medians and ltitudes 1. Write an equation in standard form of the median from in ΔC, given the coordinates as follows: (2, 1), (-1, 3), C (6, 3). 1) Find the coordinates of the midpoint of C 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 ΔC in part I. 1) Find the slope of C 2) Find the slope of the altitude, i.e. find the slope of a line perpendicular to C. 3) Write an equation of the line through having the slope found in (2). 3. Given ΔC with (4, 4), (6, 2), and C (-2, -4), write an equation of the median from. 4. Given ΔC with (-1, 5), (-7, -3), and C (5, 1), write an equation of the altitude from to C. 5. Given ΔC with (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). Write an equation of the median from E. 5
6 5.4 The triangle midsegment theorem The midsegment of a triangle is a segment joining the midpoints of two sides of a triangle. Properties of a midsegment: midsegment 1. midsegment is to the third side 2. midsegment 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 M Z P N Y Example 1) Given DE, DF, and FE are the lengths of Example 2) Given C = 42, C = 46, mid-segments. Find the perimeter of triangle C. = 48, D, E, and F are midpoints Find the perimeter of triangle DEF Example 3) D and E are midpoints. Find m< and Example 4) Points, D, and F are midpoints. m< ED. EC = 30 and DF = 23. Find C. 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! Theorem: In a, the shortest side is opposite the smallest. Theorem: In a, the longest side is opposite the largest. 52 C F 11mm 1. In C 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 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angles are NOT congruent, then the longer third side is across from the larger included angle. (converse also works) Example In each figure, name the shortest and longest segment. 7
8 Practice Problems Compare the given measures. 1. m K and m M 2. and DE 3. QR and ST Find the range of values for x
9 POINT CIRCUMCENTER POINT OF CONCURRENCENCY OF: ISECTORS OF SIDES LOCTION CUTE - INTERIOR OF RIGHT -MIDPOINT OF THE HYPOTENUSE EQUIDISTNT FROM THE VERTICES- CENTER OF CIRCUMCIRCLE OTUSE - EXTERIOR OF INCENTER ISECTORS INTERIOR EQUIDISTNT FROM THE SIDES - CENTER OF INCIRCLE ORTHOCENTER LTITUDES CUTE - INTERIOR OF RIGHT -VERTEX OF RIGHT OTUSE - EXTERIOR OF CENTROID MEDINS INTERIOR CENTER OF LNCE OR MSS 2/3, 1/3 theorem 9
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