Lesson 5 1 Perpendiculars and Bisectors Today you

Lesson 5. 1 Perpendiculars and Bisectors Today, you will learn to… > use properties of bisectors

Geometer’s Sketchpad

Theorem 5. 1 & 5. 2 Perpendicular Bisector Theorem & Converse A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment.

The distance from a point to a line is … the length of the perpendicular segment from that point to the line.

Geometer’s Sketchpad

Theorem 5. 3 & 5. 4 Angle Bisector Theorem & Converse A point is on the angle bisector if and only if it is equidistant from the two sides of the angle.

1) What do you know … about Q? T Q is a midpoint about PQ? It is a bisector about T? 7 7 C It is on the bisector about CP and DP? P is equidistant from C and D Q P D

2) Find CB. A 13 D C B CB = 13

3) Find x. 6 x + 1 = 4 4 K +1 J 6 x M L x=½

4) Find XY. Z 5. 5 Y XY = 5. 5 W X

x=6 5) Find x. Q 3 x + 2 = 20 (3 x + 2) R 20 P S

Lesson 5. 2 Bisectors of a Triangle Today, you will learn to… > use properties of bisectors and bisectors in a triangle

3 or more lines that intersect at the same point are called concurrent lines. They intersect at the point of concurrency.

A perpendicular bisector of a triangle is perpendicular to a side at the midpoint. The bisectors intersect at the circumcenter.

Perpendicular Bisectors of a Triangle Where is the circumcenter? inside the acute triangle obtuse triangle outside the on the hypotenuse right triangle

Theorem 5. 5 The circumcenter is equidistant from the vertices of the triangle.

Application… Find the circumcenter!

E GA = 13 BD = 12 F D 132 = x 2 + 52

92 = n 2 + 72 PX = 9 PB = 4. 12 XY =11. 3 B C A 92 = n 2 + 82

The State Friday, August 4, 2006 midpoints? bisectors? circumcenter? Possible point of origin & area of interest

Angle Bisectors The point of concurrency of the 3 angle bisectors is the incenter.

Angle Bisectors of a Triangle Where is the incenter? acute triangle inside the obtuse triangle inside the right triangle inside the

Theorem 5. 6 The incenter is equidistant from the sides of the triangle.

A B C

Application… Find the incenter!

2 x DG = 5 AG =13 GE = 5 = 2 12 + 2 5

NP = 8 NZ = 6 PM = 8 102 = a 2 + 82

Project? Perpendicular Bisectors Angle Bisectors

Lesson 5. 3 Medians and Altitudes of a Triangle Students need rulers & card stock to draw triangles Today, you will learn to… > use properties of medians and of altitudes of a triangle

A median is a segment connecting a vertex of the triangle and the midpoint of the opposite side. The point of concurrency for the medians is the centroid.

Medians of a Triangle Where is the centroid? acute triangle inside the obtuse triangle inside the right triangle inside the

The centroid of a triangle can be used as its balancing point. Construct a triangle and its medians.

New Project Idea!!! Equilateral, Equiangular Right Isosceles Obtuse Isosceles Acute Isosceles Right Scalene Obtuse Scalene Acute Scalene

AC = 2 cm AB = 4 cm BC = 6 cm AD = 4 cm AF = 3 cm AG = 6 cm AE = 8 cm ED = 12 cm GF= 9 cm B F E A C D G

Intersection of Medians of a Triangle 1 x = midpoint to centroid 2 x = centroid to vertex 3 x = length of median

1. Point A is a centroid. 1 x = 10 2 x = 20 3 x = 30 10 20 ED = 30 10 and AE = ___ 20 AD = ___

2. Point A is a centroid. 1 x = 7 2 x = 14 3 x = 21 7 14 FA = 7 21 AG = ___ 14 and FG = ___

3. Point A is a centroid. 1 x = 12 2 x = 24 3 x = 36 24 12 AB = 24 12 and CB = ___ 36 AC = ___

4. Find the coordinates of the centroid P of ΔDEF if D (2, 3), E (8, 5), and F (4, 1). GSP Write these coordinates down. 2+8+4 3 (4 , 2/ 3 3+5+1 3 , 3)

5. Find the coordinates of the centroid P of ΔDEF if D (-1, 8), E (-5, 2), and F (-6, -4). -1 + -5 + -6 , 8 + 2 + -4 3 3 ( -4 , 2 )

An altitude of a triangle is a segment from a vertex that is perpendicular to the opposite side. The altitudes intersect at the orthocenter.

Altitudes of a Triangle Where is the orthocenter? acute triangle inside the obtuse triangle outside the right triangle vertex of the right

Project? Medians Altitudes

6. Which special segment is AD? A median B D C

7. Which special segment is AD? A median altitude B C D bisector

8. Which special segment is AD? A altitude B D C

9. Which special segment is AD? A angle bisector B D C

Lesson 5. 4 Midsegment Theorem Today, you will learn to… > identify the midsegment of a triangle > use properties of midsegments of a triangle

A midsegment is a segment that connects the midpoints of two sides.

1. Worksheet Problem GSP

Theorem 5. 9 Midsegment Theorem The midsegment of a triangle is parallel to the 3 rd side and ___________ is half as long as the 3 rd side ____________.

2. JK and KL are midsegments of ΔABC. Find JK and AB. B JK = 10 AB = 24 J K 12 A L 20 C

3. RS, ST, and RT are midsegments in XYZ. Find the perimeter of XYZ. T Z XY = 14 7 ZY = 12 S 6 X 5 R XZ = 10 Y 36

4. JK , KL, and JL are midsegments. Find KL. KL = 2 x – 2 AB = 12 x – 36 2(2 x – 2) = 12 x – 36 B J x= 4 K KL = 6 A L C

5. What are the coordinates of the vertices of the original triangle? How does the distance formula compare to the slope formula? (Next slide)

Lesson 5. 5 Inequalities in One Triangle Today, we will learn to… > use triangle inequalities

Investigate the side lengths and the angle measures of triangles. GSP

Theorems 5. 10 & 5. 11 longest side of a triangle The ______ largest angle is opposite the ______, and the ______ shortest side is smallest angle opposite the _______.

1. List the sides in order from shortest to longest. m A = 23 m B = 111 m C = 46 First, put s in order A, C, B BC , AB , AC

2. List the angles in order from smallest to largest. U 7 T 7, 10, 11 S, T, U 10 11 S

3. List the sides of XYZ in order from shortest to longest. m X = 60 m Y = 50 m Z = 70 Y, X, Z XZ , YZ , XY

4. List the angles of XYZ in order from smallest to largest. XY = 10 YZ = 8 XZ = 6 XZ , YZ , XY Y, X, Z

How do you know if 3 segments can even be arranged to create a triangle? I had a dream…

Theorem 5. 13 Triangle Inequality In a triangle, the longest side must be less than the _____ sum of the other two sides. L<M+S

Can the side lengths form a triangle? 6. 6, 6, 15 15 < 6 + 6 No Δ 7. 6, 8, 14 14 < 8 + 6 No Δ 8. 7, 10, 13 13 < 10 + 7 Yes!

9. A triangle has one side of 8 cm and another of 17 cm. Describe the possible lengths of the third side. x could be the shortest side x could be between 8 and 17 x could be the longest side 17 < 8 + x 9<x 17 < x + 8 9<x 17, 8, x 17, x, 8 x, 17, 8 x < 17 + 8 x < 25

10. A triangle has one side of 4 cm and another of 9 cm. Describe the possible lengths of the third side. x could be the shortest side x could be between 4 and 9 x could be the longest side 9<4+x 5<x 9<x+4 5<x 9, 4, x 9, x, 4 x, 9, 4 x<9+4 x < 13

Two sides of a triangle are given. Describe the possible lengths of the third side. 12 inches and 7 inches 3<x<7 5 < x < 19 13. 10 miles and 18 miles 8 < x < 28 11. 2 cm and 5 cm

You are given a 20 -inch piece of wire. You want to bend the wire to form a triangle so that the length of each side is a whole number. 14. Identify the possible isosceles triangles that can be created. 1, 1, 18 2, 2, 16 3, 3, 14 4, 4, 12 5, 5, 10 7, 7, 6 8, 8, 4 6, 6, 8 9, 9, 2

Lesson 5. 6 Inequalities in Two Triangles Today, we will learn to… > use triangle inequalities > apply the Hinge Theorem

Compare RT and VX. V R 80° 100° T X RT > VX

Theorem 5. 14 & 5. 15 Hinge Theorem & Converse When 2 pairs of corresponding sides are congruent, the included angle of 1 is larger than the included angle of 2 if and only if… the third side of 1 is longer than the third side of 2.

< , > , or = ? YZ ____ < BC X 70 < 80 70 55 55 Y Z 50 B A 80 50 C

Find x. 5>4 3 x - 4 > 38 3 x > 42 x > 14

Find x. 30 < 38 3 x + 2 < 12 x - 7 -9 x < -9 x>1

In ΔABC and ΔDEF, AC=DF, BC=EF , AB = 11, ED = 15, and m F = 58°. Which of the following is a possible measure for C: 45°, 58°, 80°, or 90°? E B 45˚ C 11 ? ˚ 15 A F 58˚ D

In ΔGHI and ΔJKL, GH=JK, HI=KL , GI = 9, m K = 65°, and m H = 45°. Which of the following is a possible measure for side JL: 5, 7, 8, or 11? J G 9 H 45˚ I K 65˚ 11 ? L

60 15 0 10 15 5 You and a friend leave the same airport. You fly 200 miles due north then change direction and fly W 60° N for 90 miles. Your friend flies 200 miles due south then changes direction and flies E 15° S for 90 miles. I am further from the airport.

You and a friend leave from school and head in opposite directions. You drive 5 miles east, then E 30° N for 3 miles. Your friend heads 5 miles west then W 40° S for 3 miles. You both drive 8 miles, but who is farther from school? 150 30 40 140 I am further from school.