This Packet Belongs to Student Name Topic 3

This Packet Belongs to ____________ (Student Name) Topic 3: Lines, Angles, and Triangles (Part B) Unit 2– Lines Angles and Triangles Module 8: Special Segments in Triangles 8. 1 Perpendicular Bisectors of Triangles 8. 2 Angle Bisectors of Triangles 8. 3 Medians and Altitudes of Triangles 8. 4 Midsegments of Triangles Please follow along with notes. At the end of quarter 2 there will be a BINDER CHECK to check Topic 1, 2, 3, etc… 1

Objectives Apply properties of Triangle Perpendicular Bisectors Vocabulary • Concurrent • Point of Concurrency • Circumcenter 2

Prior Knowledge Needed Circum- means around _____

Perpendicular Bisectors of a Triangle Perpendicular Bisector of a Triangle: A segment is a perpendicular bisector of a triangle if it is the perpendicular bisector of a side of the triangle. Every triangle has 3 perpendicular bisectors. 4

Perpendicular Bisectors of a Triangle 5

Concurrent • Three or more lines are Concurrent if they intersect at the same point. • The point of intersection is called the Point of Concurrency. 6

Circumcenter Theorem • The 3 perpendicular bisectors of any triangle will intersect at a point that is equidistant from the vertices of the triangle. • This point is called the circumcenter and is the center of a circle that contains all 3 vertices of the triangle. 7

1) Label the GIVEN information 8

1) Label the GIVEN information 136 9

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Steps to find the Circumcenter in Coordinate plane 1) Graph Use the points given to Graph the Triangle 2) Find Perpendicular bisector Find the equation of BOTH perpendicular bisectors of the TWO legs of the triangle 3) Point of Intersection Find the point of the intersection both lines found in step 1 and step 2. • Use substitution if necessary. 11

B C 12

A B C 13

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8. 1 Classwork PAGE 365 • GO ONLINE and complete 8. 1 hw. • Alternative: Honors: 1, 5, 7, 9 -12, 15, 16 Regular: 1, 5 -6, 9 , 11, 16 Reminders: q … 15

Objectives Compare and contrast Circumscribed and inscribed circles Vocabulary Inscribed Incenter 16

Angle Bisectors of Triangles A segment is an angle bisector of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is any other point on the triangle such that the segment bisects an angle of the triangle. Every triangle has 3 angle bisectors which will always intersect in the same point - the incenter, which is • the same distance from all 3 sides of the triangle. • also the center of a circle that will intersect each side of the triangle in exactly one point. 17

Angle Bisectors of Triangles To do: draw the three angle bisectors: 18

Using the Angle Bisector of a Triangle 1. Draw/Label Given information 2) Set up an equation and solve R T ? S U No.

Angle Bisector Theorem If D is on the bisector of ∠ABC, then X A DX = DY. D B Y C The converse if ALSO true: _________ if DX=DY, then D ____________________ is on the angle bisector 20

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Using the Angle Bisector of a Triangle N ANS: Draw your diagram T 3 x+8 B W 22

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CLICK ME 24

Incenter Theorem 25

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WRAP-UP ACTIVITY Pretty Cats Are Inside Perpendicular bisector Circumcenter, Angle bisector Incenter 28

8. 2 Classwork PAGE 375 • GO ONLINE and complete 8. 2 HW • Alternative: Honors: 2 -3, 6, 8, 9, 10, 12, 14, 16, 17 -11, 21. • Regular: 2, 6, 8, 9, 12, 14, 16. Reminders: q … 29

Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle. Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle 30

Warm Up 1. What is the name of the point where the angle bisectors of a triangle intersect? Find the midpoint of the segment with the given endpoints. 2. (– 1, 6) and (3, 0) (1, 3) 3. (– 7, 2) and (– 3, – 8) (– 5, – 3) incenter 4. Write an equation of the line containing the points (3, 1) and (2, 10) in pointslope form. 32

What is the Median of a Triangle? • A median of a triangle is the line segment which MIDDLE joins a vertex to the midpoint of the opposite side. B F When you combine ALL the medians of one triangle, you get the _____. CENTROID D A E C A centroid would balance the triangle if you held it up with a pencil 33

centroid 34

Finding the Median of a Triangle Determine the coordinates of J so that SJ is a median of the triangle. Ans: Use the midpoint formula for GB J = J ( 9, 0. 5) J = 35

Centroid Theorem B X A 2 Centroid Y XC = Z of XZ XD is HALF of DC D 4 ZD is TWICE of XD C XD = DC = of XC of XZ 36

Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find LS. Centroid Thm. Substitute 21 for RL. LS = 14 Simplify. 37

Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find NQ. Centroid Thm. NS + SQ = NQ Seg. Add. Post. Substitute Subtract NQ for NS. from both sides. Substitute 4 for SQ. 12 = NQ Multiply both sides by 3. 38

Step 1. Draw/Label the given. 9 7. 2 39

Your Turn! 1. In ∆JKL, ZW = 7, and LX = 8. 1. Find KW. Centroid Thm. Substitute 7 for ZW. KW = 21 Multiply both sides by 3. 2. In ∆JKL, ZW = 7, and LX = 8. 1. Find LZ. Centroid Thm. Substitute 8. 1 for LX. LZ = 5. 4 Simplify. 40

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Example 2: Problem-Solving Application A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance? 1 Understand the Problem The answer will be the coordinates of the centroid of the triangle. The important information is the location of the vertices, A(6, 6), B(10, 7), and C(8, 2). 2 Make a Plan The centroid of the triangle is the point of intersection of the three medians. So write the equations for two medians and find their point of intersection.

Example 2 Continued 3 Solve 4 Look Back

Finding the Centroid of a Triangle Find the coordinates of the centroid of JKL. SOLUTION J (7, 10) (3, 6) N L P (5, 2) K 44

Your Turn! Find the average of the x-coordinates and the average of the ycoordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle.

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What is the Altitude of a Triangle? • Connects the vertex to where it is perpendicular to the opposite side • Every triangle has 3 altitudes that will always intersect in the same point. A T Combine all three ALTITUDE’S in a triangle and you get an ORTHOCENTER Z W R Tells us the segment is perpendicular P 47

EXAMPLE 3 ORTHOCENTERS Acute triangle Right triangle Obtuse triangle P is inside triangle. P is on triangle. P is outside triangle. Helpful Hint The height of a triangle is the length of an altitude. 48

Using the Altitude of a Triangle But wait…. . Re-read the question 49

Using the Altitude of a Triangle Step 1: Label/draw what is given Z A 17 x X W 9 x + 38 Y C Step 3: Answer the question What else do we know? 50

Using the Bisector of a Triangle In ΔABC below, AB ≅ BC and AD bisects ∠BAC. If the length of BD is 3(x + 2) units and BC = 42 units, what is the value of x? A. B. C. D. 5 6 12 13 A B D C 51

Steps to find the Ortho. Center 1) Find Equation 1 Choose a vertex • Find the equation of altitude (perpendicular to This Photo by Unknown Author is licensed under CC BY-SA leg across from vertex) 3) Point of Intersection 2) Find Equation 2 Choose a different vertex • Find the equation of altitude (perpendicular to leg across from vertex) Find the point of the intersection both lines found in step 1 and step 2. • Use substitution if necessary. Remember. If acute, Ortho is inside triangle. If its right triangle, its on the triangle, and if its obtuse, ortho is outside.

Finding the Orthocenter Find the orthocenter of ∆XYZ with vertices X(3, – 2), Y(3, 6), and Z(7, 1). Step 1 Graph the triangle. X

Example 1 Continued Point-slope form. Substitute 6 for y 1, and 3 for x 1. Distribute . Add 6 to both sides. for m,

Example 1 Continued Step 4 Solve the system to find the coordinates of the orthocenter. Substitute 1 for y. Subtract 10 from both sides. 6. 75 = x Multiply both sides by The coordinates of the orthocenter are (6. 75, 1).

Example 2

Example 2 Continued Notice that this Orthocenter is OUTSIDE the triangle

Checking the Orthocenter Show that the altitude to JK passes through the orthocenter of ∆JKL. 4=1+3 4=4 Therefore, this altitude passes through the orthocenter.

WRAP-UP EXAMPLES Name all segments that are (if any) • Angle Bisectors QU • Perpendicular Bisectors NONE • Altitudes RT • Medians SP Pretty Cats Are Inside Perpendicular bisector Circumcenter, Angle bisector Incenter Mean Cats Are Outside Median Centroid Altitudes Orthocenter 60

Lesson Quiz Use the figure for Items 1– 3. In ∆ABC, AE = 12, DG = 7, and BG = 9. Find each length. 1. AG 8 2. GC 14 3. BF 13. 5 For Items 4 and 5, use ∆MNP with vertices M (– 4, – 2), N (6, – 2) , and P (– 2, 10). Find the coordinates of each point. 4. the centroid (0, 2) 5. the orthocenter

8. 3 Classwork Page 390 • GO ONLINE and complete 8. 3 HW • Alternative: Honors: 1, 4, 5, 7, 8, 11, 12, 13 -20, 23 • Regular: 1, 4, 7, 8, 12, 13, 17, 23 Reminders: q … 62

Objectives Apply properties of midsegments Vocabulary Midsegment of a triangle 63

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Triangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and is half as long. 65

Key Concepts, continued • Every triangle has three midsegments. 66

Common Errors/Misconceptions • assuming a segment that is parallel to the third side of a triangle is a midsegment • incorrectly writing and solving equations to determine lengths • incorrectly calculating slope • incorrectly applying the Triangle Midsegment Theorem to solve problems • misidentifying or leaving out theorems, postulates, or definitions when writing proofs 67

QUICK PRACTICE 68

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Using the Triangle Midsegment Find the lengths of BC and YZ and the measure of ∠AXZ. 1. Identify the known information. 71

Using the Triangle Midsegment (cont. ) Triangle Midsegment Theorem Substitute 4. 8 for XZ. Solve for BC. 72

Using the Triangle Midsegment (cont. ) Triangle Midsegment Theorem Substitute 11. 5 for AB. Solve for YZ. 73

Using the Triangle Midsegment (cont. ) 4. Calculate the measure of ∠AXZ. Triangle Midsegment Theorem Alternate Interior Angles Theorem 5. State the answers. BC is 9. 6 units long. YZ is 5. 75 units long. m∠AXZ is 38°. ✔ 74

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PRACTICE QUESTIONS 76

PRACTICE QUESTIONS 77

CHALLENGE!! The midpoints of a triangle are X (– 2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle. 78

8. 4 Classwork Page 399 • GO ONLINE and complete 8. 4 HW • Alternative: Honors: 2, 4, 5, 6, 8, 12, 13, 17, 20, 22, 24 • Regular: 2, 4, 6, 8, 11, 20, 22, Reminders: q … 89