Period: ___5.1 Isosceles & Equilateral Triangles an Altitude Is

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

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.

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

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 N ___ || NP P

NP || ___

3) 4) X S

P M

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Find the missing length indicated. 5) Find PQ 6) Find DC

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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.

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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.

1. Once you have placed the pieces in the square on the right and the one of the left, explain why 푎2 + 푏2 must be equal to 푐2 in terms of the area they form. Be very specific. You are writing a paragraph proof.

2. Algebraic proof of the Pythagorean Theorem: a) Use algebra to describe the area of the square on the right using the side length:

b) Describe the area of the square in terms of all the 5 pieces that make up the square. (There are 4 congruent triangles and 1 square)

c) What should be true about the area from part 2a, and from part 2b?

d) Use an equation to relating what you said is true in part 2c to show that 푎2 + 푏2 = 푐2

3. Garfield’s Proof: President Garfield proved the Pythagorean Theorem in a very similar manner to what we did in problem 2, but he used this shape on the right.

Follow a similar process to what we did in problem 2 Name:______Period: ______5.10 Pythagorean Theorem

You and your partners are the lead project designers for a large company. You are developing new technology that may allow drones to deliver packages. You are currently testing the software on the drones, and you are verifying that they can make several deliveries in one trip.

Part a: Use the distances of each street intersection to determine the total length of the flight path your drone will take (marked with the dotted line). The drone starts at the lower left of the map. The map is on the reverse. Round to two decimal places. Please label the side lengths of each right triangle.

Part b: Your drone can fly at 30 feet per second. If it stops at each delivery site (marked with an X) for 2 minutes. How long (in minutes) will it take for the drone to complete the deliveries AND return home? Show your calculations

Part c: A car (which must travel using the streets) can average about 25 miles per hour on the surface streets of Tucson. The vehicle will stop at each delivery location for about 2 minutes and 30 seconds. In addition the car will have to wait about 20 seconds at each intersection due to red lights. Calculate the time it would take a car to make the same delivery route

The map: All units are in feet and indicate the distance between streets. Assume all street corners form right angles and run parallel to each other.

657 ft 657 ft

534 ft

Name: ______Period: ______5.11 Pythagorean spirals

After reviewing the video, start making your Pythagorean spiral on this paper. Use the spirals on the right and left to guide you in your construction. You need a protractor and a ruler. The edge segments of your spiral should be 1” Use the point provided as the center of your spiral. You should make 14 triangles and you should color them or place the spiral in a picture/scene, how are you inspired?

Hyp. Length

Number 퐻1

퐻2 퐻3

퐻4

퐻5

퐻6

퐻7 퐻 8 퐻9

퐻10

퐻11

퐻12 퐻 13 퐻14

Video source: http://www.youtube.com/watch?v=b3JafYAW1wg Once you have made your spirals, turn the paper over and answer the questions on the back 1. Label each hypotenuse 퐻1, 퐻2, 퐻3 and so on, where 퐻1 is the smallest hypotenuse you first constructed. Starting at 퐻1 find its length. Then go find the lengths of 퐻2, 퐻3, … until you have found the length of 퐻14. Use the table on the front side to save your answers there. What pattern can use to find the length of the 513th hypotenuse? What would that length be approximately?

1 2. Starting with the first triangle you constructed, find its area. Remember the area of a triangle is 퐴 = (푏푎푠푒)(ℎ푒𝑖𝑔ℎ푡). Find the area of the 2nd 2 triangle, and the 3rd, 4th, and so on. What pattern could you use to find the area of the nth triangle? Write the formula for its area.

Name: ______Period: ______5.12 Special Right Triangles

1) Each triangle above is an isosceles right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse in simplified radical form.

2) How can you calculate the length of the hypotenuse of an isosceles right triangle if you know the length of the legs?

3) Each triangle below is an isosceles right triangle. Use the Pythagorean Theorem to find the length of the legs in simplified radical form.

105

4) If you know the length of the hypotenuse, how would you find the length of the legs?

5) Find the missing side lengths 5√2

18 b 14√2 a = ______b = ______c

a c = ______

5) In the space above, use a compass and straight edge to construct a large equilateral triangle ΔABC. What are the measures of each angle ∠A, ∠B and ∠C? ______

6) Construct the perpendicular bisector of 퐴퐵 through point C, label the point D where the perpendicular bisector intersects 퐴퐵. What are the measure of ∠ADC and ∠BDC? ______What is the measurement of ∠ACD and ∠BCD? ______How do you know? ______7) Is ΔADC ≅ΔBDC? ______If so, by which triangle congruence? ______

8) Is 퐴퐷 ≅ 퐵퐷? How do you know? ______

9) How do 퐴퐶 푎푛푑 퐴퐷 compare? ______Will that always be true? ______10) To the right are two perpendicular lines. Place point M on the horizontal line. Measure a 600 angle from M to the vertical line (extend if needed). Label that point P. You just made a 30-60-90 triangle.

In cm measure 푀푁 ______and 푀푃 ______푁푃 ______How does the shorter side compare to the longer side of the triangle?

11) Use the Pythagorean Theorem to find the hypotenuse 푀푃 12) Using the ratio of a 30-60-90 triangle, find the missing side lengths.

g g = _____ 8 f = _____ d e e = ______f h = __ d = ______5 TVHS Geometry Name______©z b2d0[1M6u pKBuBt^aL FSIonfptlwYaFrCe\ mLlLuCd.x v DAyltlx krBiKgfhxtDst \rTeksvevrkvPeWdL. 5.13 Practice with Special Triangles Date______Period____ Find all missing side lengths. Leave your answers as radicals in simplest form. Show your work. 1) 2) 2 n 60° n m 45° m 6

3) 4) a b u 3 2 60° 2 45° 6 v

5) 6)

m 30° n x y 45° 3

3 2

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6) How do you construct the point of concurrency in question 5?

1) In ∆ABC, what point of concurrency is O? 7) Describe the construction. What is M? 2) O is equal distance from what part of the triangle? 8) Describe 퐶퐷.

9) Find the length of WY (diagram above).

Point G is the incenter of ∆ABC

3) How is it created?

4) Point G is equal distance from what parts of 10) Given 푚∠푄푆푅 = 푚∠푄푆푇 푎푛푑 푇푄 = the triangle? 1.3 푓푖푛푑 푅푄 (diagram above)

11) PQ is the perpendicular bisector or ST. Find the values of m and n

5) This young man is balancing the triangle on its center of mass. What point is that called? 12) 푆푁 푇푁 푎푛푑 푉푁 are perpendicular bisectors ∆PQR. Find NR 20) Solve for y. RV TR y QN

21) Solve for x 13) Solve for n (above diagram)

22) What is the perimeter of the triangle?

14) Solve for n (above diagram) 23) What is the area of the triangle?

24) Solve for x

15) 푃푄 is midsegment of ∆RST. What is length of 푅푇 ? 25) Solve for x

16) Can 4, 7, 10 be sides of a triangle?

17) Simplify −3√24푥3 26) Find x and y 18) Simplify √12 + 2√48

19) Solve for x

5.15 Geogebra Construction Lab

Your work will be submitted to your teacher in a word document or a google doc via email. You will copy and paste the images of your work from each activity into that file and then send/share it to your teacher. For the subject line use “Geogebra Lab” Geogebra Link: www.geogebra.org Activity 1: To construct an Equilateral Triangle

1. Draw two points A and B using the New Point tool . 2. Draw the line segment AB using the Segment between Two Points

tool . 3. From A draw a circle through B using the Circle with Center through a Point tool .

4. From B draw a circle through A using the Circle with Center through a Point tool

5. Find one intersection C of the two circles using the Intersect Two Objects tool .

6. Draw the line segments AC and BC using the Segment between Two Points tool . 7. Hide the circles by right clicking them and deselecting Show Object.

Activity 2: To construct an isosceles triangle

1. Select the tool (Circle with Center through a Point) and construct a circle center A through point B. If the labels are not showing, right click, select Properties and with the Basic tab open, click on Show Label.

2. Select the New Point tool and construct any point C on the circumference of circle c.

3. Select the Segment between Two Points tool and construct [AC] 4. Construct [BC]. 5. Right click on one side of the triangle, select Properties, and with the Basic tab open, click on the drop down arrow beside the Show Label box. Select Name and Value to show the name and length of this side of the triangle. Repeat for the other triangle sides. 6. Drag each vertex of triangle ABC and note the length of its sides. 7. Hide the circle, by right clicking on it and clicking on Show Object.

8. Measure the 3 angles in the triangle using the tool. Drag any of the vertices of the triangle ABC and observe how the angle measures change.

Activity 3: Constructing Medians and constructing the Centroid of a triangle

(A median is a line segment connecting any vertex of a triangle to the midpoint of the opposite side)

1. Click on File and select New Window.

2. Draw a triangle using as above.

3. Using the ,and tools, construct the medians of each side of the triangle. 4. Construct the intersection of the medians by

selecting the tool. 5. Drag any of the vertices of the triangle and note that the 3 medians remain concurrent, at the CENTROID. Activity 4: Constructing Median and constructing the circumcenter and circumcircle of a triangle

(A median is a perpendicular bisector of a line segment)

1. Click on File, New Window, and draw a triangle using as above.

2. Select i.e. Midpoint or Center tool and selecting each side of the triangle in turn, construct the midpoints of each side. 3. Using the Perpendicular Line Bisector tool, select each side to construct perpendicular bisectors (medians) of each side.

4. Select the Intersect Two Objects tool and then 2 of the medians to construct the circumcenter. 5. The equations of the 3 medians are shown in the Algebra window. 6. Hide the medians by right clicking on each one and clicking on Show Object. Drag the vertices to see the circumcenter change position.

7. Click on the Circle through a Point tool , then the circumcenter (point of intersection of the medians) and one of the vertices of the triangle and construct the circumcircle, which passes through the 3 vertices. 8. Drag the vertices of the triangle to confirm the construction.

5.15 Geogebra Construction Lab

Activity 5: Constructing the bisectors of the angles and constructing the incenter and incircle of a triangle.

1. Construct a triangle ABC in a new window. Select , the Angle Bisector tool. Select the points B, A and C, in that order, to construct the angular bisector of

2. Select the tool and 2 of the angle bisectors to construct the incenter. 3. Hide the angle bisector lines.

4. Selecting the tool, draw a perpendicular line from the incenter D, to line AB or any of the 3 sides

of the triangle. With the tool selected construct the intersection E of side AB and this perpendicular line.

5. Hide the perpendicular line. Select , and with D as center and E as the point on the circle, construct the incircle.

6. Drag the vertices to confirm the construction.

Activity 6: Find the measurements of the interior angles of a polygon.

1. How does the method of constructing ABCD in steps 1 and 2 guarantee a quadrilateral that is a parallelogram?

2. What are two conditions that must be met for a quadrilateral to be a rectangle? Write a theorem that states the theorem.