Name: _________________________________ Period: ________ 5.1 Isosceles & Equilateral Triangles An altitude is a perpendicular segment from a vertex to the line containing the opposite side. 1. Prove: the altitude to the base of an isosceles triangle bisects the base. 2. An obelisk is a tall, thin, four sided monument that tapers to a pyramidal top. The Washington Monument on the National Mall in Washington D.C. is an obelisk. Each face of the pyramidal top is an isosceles triangle. The height of each triangle is 55.5 feet, and the base of the triangle measures 34.4 feet. Find the length, to the tenth of a foot, of one of the two equal legs of the triangle. 3. With your compass, carefully construct two circles- one with A as a center and AB as the radius, the other with B as the center and BA as the radius. Label one of their intersections as point C. Use your straight edge to construct ΔABC. What kind of triangle is ΔABC? Write a paragraph proof. Find each value. 4. 푚∠퐴 = ______ 5. 퐶퐴 = _______ 6. 7. 푓푖푛푑 퐷퐺 = ________ 푓푖푛푑 ∠푇 = ______ 8. 9. 푚∠퐷 = ______ 푡 = _____ 10. 11. 푓푖푛푑 ∠퐴퐵퐶 = ______ 푓푖푛푑 ∠퐻 = ______ 푓푖푛푑 ∠퐴퐶퐷 = ______ 푓푖푛푑 ∠퐴퐷퐶 = ______ Name: _____________________________ Period: ________ 5.2 Bisectors and circumcenters 1. Create the perpendicular bisector of 퐴퐵, and create a point 푃 on the bisector. How far is the point P from A, and how far is the point P from B? 2. Create the perpendicular bisectors of the triangles and label the circumcenter as point X. How far is point X from the vertices of the triangle? Measure them and show you’re correct. 3. Create all 3 bisectors of the triangle and show that they meet at a single point. Then circumscribe the triangle. 4. A group of astronomy students are each independently working on a project at the University of Arizona. Jim is at the college of optical sciences, Claire is at the Steward observatory, and Carl is located at the University of Arizona Library. They all plan to meet and eat lunch on a warm sunny day, but they all agree that they should all travel the same distance to meet each other. Determine the location where they should meet for lunch. 5. A radio station in Hawaii has hidden a treasure somewhere on the main island. Every day they will give a clue as to how to find their hidden treasure. The first day the clue is: The treasure is not near the coast. The second day the clue says that the treasure is located the same distance from Mauna Kea, as it is from Mauna Loa. The next day they give a clue that the treasure is 28.5 km away from the town of Mountain View. Determine the location of the treasure. Name: _______________________________ Period______ 5.3 Incenter and ∠ bisectors 1) Bisect ∠푃 angle with ray 푃퐶⃗⃗⃗⃗⃗ , show your construction marks. a) Label point C on the angle bisector b) Construct the perpendicular from point C to each ray of ∠푃 P c) Label the intersections B and K d) Measure 퐶퐵 푎푛푑 퐶퐾 e) What do you notice? 2) Measure ∠D a) Measure an equal distance (in cm) on each ray of ∠D. b) Label these points O and G c) Create the perpendiculars from each ray of ∠D through O and G. d) Label the intersection of the perpendiculars as point T D e) Draw 퐷푇⃗⃗⃗⃗⃗ F) Measure the angles that are created ∠ODT and ∠GDT. G) What do you notice? Incenter 3) Find the angle bisector of each angle of the triangle. Show your work. The place the angle bisectors intersect is the “incenter” and it is always INSIDE the triangle. 2) You should be able to use the incenter of the triangle to inscribe a circle inside the triangle (this means the circle is inside of the triangle, the center of the circle is the incenter of the triangle, and the edge of the circle should just touch each side of the triangle). The incenter is equal distance to each side of the triangle. Draw each inscribed circle. 3) Legend has it that a treasure ship sank equidistant from the routes that create the Bermuda Triangle. Use the map below, show all construction marks, and locate where the sunken treasure lies. Name: ____________________________Period: ________ 5.4 Medians and Centroids Median of a triangle is a segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side. A 1) Find the median from vertex A B C CENTROID : the point of concurrency of the medians of the triangle 2) Find the median of each side of the triangle. Label the centroid as point P. Show your work. B A E C F 3) True or False: The point at which a triangular table could balance on one leg is the same distance from each side of the table. Name: ____________________________Period: ________ 5.4 Medians and Centroids P I In the space below do the following constructions: 1) Construct a large triangle ΔRST, use compass & straight edge 2) Construct the Circumcenter, label it A 3) Construct the Incenter, label it B 4) Construct the Centroid, label it C Geometry Name___________________________________ ID: 1 ©T M2K0k1w4Q FK_u\tBaX JSooif\tFwTaArJeS VLdLWCJ.C j RALlNlt jrUijgbhhtwsP PryeGsMeOrGvke_dP. 5.5 Midsegments Mania Period____ In each triangle, M, N, and P are the midpoints of the sides. Name a segment parallel to the one given. 1) 2) I M H E M D P N G P N ___ || NP C NP || ___ 3) 4) X S P M M N V N W XV || ___ T P R MN || ___ Find the missing length indicated. 5) Find PQ 6) Find DC E T D C 24 P S 14 U G Q F Worksheet by Kuta Software LLC ©s I2U0I1l4y IKKuBtNaW gSBoFfMtpw_aBrYeb DLVL_CZ.V h `AplVlF FrPilgmhFtpsI Uryeas[eVrqvwerd-w1.J- E UMYaTdHe` [wWiNtxhy _IVnpf^iWnsiMtweR qG[eKoumbeXt^rVy]. 7) Find KL 8) Find WY S X F G L 12 20 W Y R K Q Solve for x. 9) 10) F M Z Y x + 33 x - 5 G x + 21 E x + 2 G K H L 11) 12) M D C D I J 2x - 2 2x - 12 N C E L 2x + 8 12 + x Find the missing length indicated. 13) Find LN 14) Find FH N H Q x + 11 2x + 19 W 2x - 4 G R F 2x - 12 M V L Worksheet by Kuta Software LLC ©a u2d0a1[4m tKEu_tXaM HSvobfptAwZaHrOea QLwLfCY.E o GASlSle SrXifgLhXtNs_ drqeLskesrUv^ek-d2w.-f t nMMaJdaew LwDiJtRhP aINnMfkionRi[tMe^ uGBeUozmBept`rwyu. Name: ___________________________ Period: ______ 5.7 Triangle Properties Quiz Review 1. Match the statements in the table with the words/phrases contained in the word bank Word Bank: Center of Mass Inscribed Angle Bisectors Circumscribed Perpendicular Bisectors Equidistant from Vertices Medians No circle Equidistant from Sides Circumcenter Incenter Centroid Formed by the intersection of: Type of circle Special property A 2. Using ∆퐴퐵퐶 find: a. The value of x 3x + 15 BA=24 b. Find the value of y 푚∠퐶 = (2푦 + 12)° B C 3. Find the value of x 4. Find the value of x 5. Find the length of 푍푌 and 푆푄 6. 푍푌 = 푆푄 = 8. Oscar wants to open a restaurant, but he is concerned with the competition he faces from 3 competitors. If Oscar wants to build his restaurant an equal distance away from his 3 competitors. Find the location which he should establish his business. Show all construction marks. Explain your reasoning for why you constructed that location. 9. The safety department at a regional airport has decided to building to house an emergency response team who will quickly get to the runway to deal with a crashed aircraft. The inspector decides the building should be the same distance from all the runways so that emergency vehicles can quickly get to any of the three runways no matter where a plane may crash. Determine the location of where to build the building and where to lay pavement for the vehicles to get to each runway. Explain your reasoning for why you constructed that location. :%;< %9%')%. ( =;>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2 1 ;@ $A@') B6('3' 7#8 A <2$%1'&(<1 /(A,1) !>>>>>>>>>>>>>>>>?% )>>>> D D ED FD D GD HD :D ID D 7 8(49 !" # $"%&'%%() #(*!+,-.)/ C C 0)1%%/21$3%/+%30*#* '!445'6 / D D ED FD D GD HD :D ID D 7 8(49 2$; = 3 *)6 / *4!3 * !<# %&'+>()2&(B>,;)CC %)0)8)"'#;%0// 24%%5<*# !'*B402< Use the square on the left to draw a shape similar to the image (right). The goal is to produce 4 right triangles with sides a & b with a hypotenuse c. Sides a and b should be different lengths. Do this by marking off equal distances on every side of the square so that a square with side lengths of c are formed inside. Transfer the exact same drawing to the square on the left below. Once you have drawn it for a second time, please cut out this top portion and cut out your 4 right triangles and the square. Name: __________________________ Period: _______ 5.9 Proving The Pythagorean Theorem Redraw the triangle from above here. Assemble the cut out pieces here. Using the 4 cut out triangle pieces, assemble them in the square on the right so that the larger square on the right has 2 squares of different sizes in it with side lengths of a & b.