Geometry P/AP. January 8 22, 2018 TRIANGLE PROPERTIES Date Topic Assignment

Transcription

1 Geometry P/P. January 8, 018 TRINGLE PROPERTIES ate Topic ssignment Monday 1/ Mid-Segment Theorem in Triangles. T Pg 04, # 1-3 Tuesday 1/09 Wednesday 1/10 Thursday 1/11 Friday 1/1 5-3 Perpendicular and ngle bisectors. Pg. 11. # isectors in Triangles. ngle and Perp. isectors. Pg. 17: 1-, Ws 5-5 isectors in Triangles. Medians and ltitudes. Pg. : 1-4, Ws 5-6. Indirect Proofs. Quiz 1: 5- & 5-3 Pg. 7: #1-1-1, Mon. 1/15 ML KING HOLIY Tuesday 5-7. Inequalities in One Triangle. 1/16 Tuesday 1/17 Inequalities in Rt. Triangles. WS -Pythagorean Theorem more or less. Quiz : 5-4 & 5-7 WS Inequalities in One Triangle WS -Pythagorean Theorem more or less Tuesday 1/18 Friday 1/ Inequalities in Two Triangles (Hinge Theorem). Review: 5-1 to 5-8. WS Inequalities in Two Triangles T. Pg. 40: #1-45, Pg. 44: 3, 9, 16, Study for Test. Monday 1/ Tuesday 1/3 Wednesday 1/4 Thursday 1/5 Test 1: 5- to Pythagorean Theorem, and its onverse WS: 8. Special Right Triangles Page 48: 15-1, 3, 5, Special Right Triangles. Worksheet on right triangles 8-. pplications of Pythagorean and Special Right Triangles to solve problems. WS-Right Triangle pplications

2 5-1 Midsegments of Triangles (Notes) midsegment of a triangle is a segment connecting the midpoints of two sides. Triangle Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. One way to prove the Triangle Midsegment Theorem is to use similar triangles. nother way is to use coordinate geometry and algebra. This style of proof is called a coordinate proof. Given: TRI where T(0,0) R(6, 10) I(8, 0) is the midpoint of TR F is the midpoint of TI Verify: F RI and F = ½ RI Sample Problems: 1. M, N, and P are midpoints; MP = ; MN = 4; perimeter of MNP = 60. Find NP and YZ M P Y N Z

3 . Find m MN and m NM. N M is a new bridge being built over a lake as shown. Find the length of the bridge. 963 ft 640 ft 963 ft 4. The perimeter of a triangle is 78 ft. Find the perimeter of the triangle formed by its midpoints. 5. Only one of the lengths a, b, or c can be found. Name the segment and find its length. 4 3 b 4 3 a c E = x + 5 and = 3x 1. Find the value of x E 14

4 5.3 oncurrent lines When 3 or more Lines intersect in one point, they are. Perpendicular isectors concurrent at the circumcenter which is equidistant from the vertices Ex. Find the center of the circle that circumscribes Δ YZ. (1, 1) Y (1, 7) Z (5, 1) y ngle isectors concurrent at the incenter which is equidistant from the sides of the triangle Ex. ity planners want to locate a fountain equidistant from 3 straight roads that enclose the park. Explain how they can find the location. x Ex. The towns of damsville, rooksville, and artersville want to build a library that is equidistant from the 3 towns. Show where they should build it. Medians concurrent at the centroid (also called the center of gravity) W Ex. M is centroid of Δ WOR and WM = 16. Find W. Z M Y O R

5 ltitudes concurrent at the orthocenter K Ex. Is K a median, an altitude, neither or both? L M Ex. is the centroid of Δ EF. If GF = 6x +9y, write an expression that represents F. G F H E Ex. Given (3, 3) and (5, 7), find points on the perpendicular bisector of. Verify your results by showing each point is equidistant from and. y x Ex. (6, 8) O (0, 0) (10, 0) a. Write equations of lines k and m such that k O at and m O at. b. Find the intersection of lines k and m. y x

6 Geometry Section 5.3 Name ate Period Medians, ltitudes, Perpendicular isectors Four special types of segments are associated with triangles. median is a segment that connects a vertex of a triangle to the midpoint of the opposite side. n altitude is a segment that has one endpoint at a vertex of a triangle and the other endpoint on the line containing the opposite side so that the altitude is perpendicular to that line. n angle bisector of a triangle is a segment that bisects an angle of the triangle and has one endpoint at the vertex of that angle and the other endpoint on the side opposite that vertex. perpendicular bisector is a segment or line that passes through the midpoint of a side and is perpendicular to that side. P omplete using the figure at the right. 1. If =, then is a median of P.. If P is a perpendicular bisector of then =. 3. If P is a right angle, then and are altitudes of P. 4. If P is a median of P, then =. 5. If = and P, then is a perpendicular bisector of. 6. If P and are both altitudes of P, then is a right angle. omplete. E 7. If bisects, then and =. G 8. If is the perpendicular bisector of E, then E = and E =. 9. If = F, then is the perpendicular bisector of F, and F. 10. If = G, then is the bisector of. F

7 omplete the statement. 11. If is on the bisector of SKN, then is equidistant from and S K 1. If is on the bisector of SNK, then is equidistant from and. N 13. If is equidistant from SK and SN, then lies on the. 14. If O is on the perpendicular bisector of L, then O is equidistant from and. O F 15. If O is on the perpendicular bisector of F, then O is equidistant from and. 16. If O is equidistant from L and F, then O lies on the. L 17. Find if is a median of. 18. Find if is an altitude of. x-7 (4x+10) x+3 x-17 x 3x Find m if is an angle bisector of. m =(4x-6) (x+6)

8 In problems 0-3, (,5), (1,-1), and (-6,8) are the vertices of. y 0. What are the coordinates of K if K is a median of? 1. What is the slope of the perpendicular bisector of? x. What is the slope of L if L is the altitude from point? 3. Point N on has coordinates 8 1, 5 5. Is N an altitude of? Explain your answer. 4. RT is a median in RL with points R(3,8), T(1,3), and (9,1) a. What are the coordinates of L? y b. Is RT an altitude of RL? x c. The graph of point S is at (4,13). S intersects R at. If is at (6,10), is S a perpendicular bisector of R?

9 Geometry Worksheet Inequalities in One Triangle Name ate Period In exercises 1-10, the lengths of two sides of a triangle are given. First, write an inequality to describe all possible values for x, the length of the third side of the triangle. Then, if domain of x is limited to the set { 1, 1, 4, 7, 9.3, 14, 19 }, list all possible values for x. 1. 7, , ,.3. 5, , , , , , , If the lengths of two sides of a triangle are 9 and 15, between what two numbers does the length of the third side lie? 1. If the lengths of two sides of a triangle are 3 and 7, then the length of the third side must be less than. 13. The lengths of two sides of a triangle are 7 and 10. etween what two numbers does the third side lie? 14. The length of a leg of an isosceles triangle is 9. etween what two numbers does the length of the third side lie? 15. Each leg of an isosceles triangle has length 1. etween what two numbers does the length of the base lie? In exercises 16-4, state whether it is possible for a triangle to exist with sides of the given lengths , 11, x, x, x (x>0). x, y, x+y (x and y > 0) 17. 9, 9, x, x, 4x (x>0) 3. x, y, x-y (x and y > 0) 18. 8, 15, 7 1. x, x+3, x+4 (x > 1) 4. x, x+, 4x+3 (x>0) 5. In, m = 40 and m = 110, what is the longest side of? 6. What angle is the largest in PQR? 7. What side is the shortest in? P 8. What side in a right triangle is the longest? 9. If >, and >, then which angle is larger, 1 or 4? If m VRT = 10 and m S > m T, then the longest side of RST is. 31. In, if, m 1 = 6x + and m = 8x 0, find m. 5 Q 6 R 1 V 3 1 R T S

10 3. Which side is the longest? 33. Which side is the longest? the shortest? the shortest? G 110 a I a H E F Which side is the longest? the shortest? S 5+3 T W If the sides of a triangle have the following lengths, find all possible values for x. 35. = x + 5, = 3x, = 4x PQ = 3x, QR = 4x 7, PR = x + 9

11 PYTHGOREN THEOREM-MORE OR LESS (1) heck to be certain that the lengths can fonn a triangre. () Let c be the longest side._ If if > a + b, then the triangle is obtuse. If cl < a + b, then the triangle is acute. lassify eacb triangle witb the given side lengths as acute, right, or obtuse. 1., 5,6 3.9,40,41 5.9, 8,6 7. 1,, Vi 9.3,3.5,4.'5,8, ,.5, 6 6. I!, 6, 3i 8.1,14, ,10,18 ecide if LI is acute, rigbt, or obtuse. II IS ~ ~----t c. 1t-8.~_t=-_~.. """"'c-...ol~. )9.~ - -. IIFG 15 ontest Problem 0. In a room 8 ft. wide, 8 ft. high, and 15 ft. long - a bug crawls from the middle oftbe front wall, 1 ft. above the floor, to the middle of the back wail, 1 ft. below the ceiling.. What is the shortest path? How far is it?

12 etermine whether it is possible to have a triangle with the given vertices. then determine if the triangle is obtuse, acute, or right? If a triangle does exist, (1) (,3), 8(-5, -11), (-8, 15) () J(1, -4), K(-3, -0), L(5, 1)

13 Worksheet Section 5.5 Inequalities in Triangles Name ate Period I. omplete each statement using sometimes, always, or never. T R S 1. If RS and ST, then is to RT.. If RS and RT, then m is = to m R. 3. If RS, RT, and R, then is to ST. 4. If RT, ST, and > RS, then m is > m T. 5. If RT, ST, and m > m T, then is > RS. 6. If m = m R, m = m S, and > RS, then is < RT. II. State whether or is the greater measure III. State whether 1 or is the larger angle

14 IV. 15. In the plane figure, = 10, =9, = 11. ompare the measures of angles 1,, and In the plane figure, m 1 = 100, m = 90, m 3 = 10. ompare the lengths of,,, and Given: > Which segment is shorter, R or R? R W 18. Given: m R > m T Which segment is longer, WT or WS? R S T 19. Given: m > m m E > m E Which segment is longer, or E? 0. Name the longest segment. a) b) W Z W Z Y Y VI. 1. Name all segments that we know to be congruent.. Which is shorter, E or E? 3. Which is longer, E or E? 4. Which is the shortest segment in the figure? E