Chapter 6.1 Medians. Geometry

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1 Chapter 6.1 Medians Identify medians of triangles Find the midpoint of a line using a compass. A median is a segment that joins a vertex of the triangle and the midpoint of the opposite side. Median AD creates two segments, BD and CD. Is BD CD? Yes How many medians does a triangle have? 3 Notice that all the medians intersect a one point, called the centroid. In Physics, we call this the center of mass. A C D B

2 Chapter 6.1 Medians Identify medians of triangles When three or more lines or segments intersect at the same point, the lines are concurrent. It can be shown that the length from the vertex to the centroid is twice the length from the centroid to the midpoint. A Theorem 6.1: The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. 2x x Bookwork: page231 problems C D B

3 Chapter 6.2 Altitudes and Perpendicular Bisectors Identify altitudes and perpendiculars of triangles Construct the altitude of a triangle. An altitude of a triangle is a perpendicular segment from a vertex to the opposite side. AD is an altitude of triangle ABC. How many altitudes can a triangle have? 3 A perpendicular bisector is a line or segment that bisects a side of a triangle. Line m is a perpendicular bisector with point E being the midpoint of CB m A Bookwork: page238 problems 8-23 C E D B

4 Chapter 6.3 Angle Bisectors of Triangles Identify angle bisectors of triangles Construct an angle bisector. An angle bisector of a triangle is a segment that bisects an angle of the triangle. 1 2 How many angle bisectors does a triangle have? 3 Notice BD does not have to be congruent to CD A 1 2 Can they be? Yes Bookwork: page 242, problems 7-20 C D B

5 Chapter 6.4 Isosceles Triangles Identify properties of isosceles triangles Is AB AC Yes, all radii of a circle are congruent. Construct the angle bisector of angle A. Construct the midpoint of BC. AD is the perpendicular bisector of angle A. A Therefore, ACD ABD B C

6 Chapter 6.4 Isosceles Triangles Identify properties of isosceles triangles In Chapter 5.1 we determined an isosceles triangle has two congruent sides. From the previous demonstration, an isosceles has two congruent angles. Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 6-3: The median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle. Converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Theorem 6-5: A triangle is equilateral if only if it is equiangular. Bookwork: page249, problems 6-21

7 Chapter 6.5 Right Triangles Use tests of congruence for right triangles. In a right triangle, the side opposite the right angle is the hypotenuse. The two sides that form the right angle are called legs. If the two legs of a right triangle are congruent to two legs of second right triangle, the triangles are congruent by SAS. This leads to the following theorem. LL Theorem: If two legs of one right triangle are congruent to the corresponding legs of a second right triangle, then the triangles are congruent. If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of a second triangle, the triangles are congruent by AAS. HA Theorem: If the hypotenuse and acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of a second right triangle, then the triangles are congruent.

8 Chapter 6.5 Right Triangles Use tests of congruence for right triangles. If a leg and an acute angle of a right triangle are congruent to the corresponding leg and acute angle of a second right triangle Triangles congruent by ASA Triangles congruent by AAS LA Theorem: If one leg and acute angle of one right triangle are congruent to the corresponding leg and acute angle of a second right triangle, then the triangles are congruent. HL Postulate: If the hypotenuse and leg of one right triangle are congruent to the corresponding hypotenuse and leg of a second right triangle, then the triangles are congruent. Bookwork: page 254, problems 7-21

9 Chapter 6.6 The Pythagorean Theorem b Define and use the Pythagorean Theorem a c A Square c b a The area of the red square is? a + b a + b = a 2 + 2ab + b 2 The area of a triangle is? 1 2 lw or 1 2 bh a c RIGHT? c b The area of all red triangles is? 1 ab 4 = 2ab 2 The area of the blue square is? a 2 + 2ab + b 2 2ab = a 2 + b 2 b a a 2 + b 2 = c 2

10 Chapter 6.6 The Pythagorean Theorem Define and use the Pythagorean Theorem The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse, (c), is equal to the sum of the squares of the lengths of the legs (a and b). a c c 2 = a 2 + b 2 b The Converse of Pythagorean Theorem: If the measure of the longest side of a triangle, c, the measure of the lengths of the other two sides, a and b, and a 2 + b 2 = c 2, then the triangle is a right triangle. Bookwork: page 260, problems 17 40; emphasis on problem 40.

11 Chapter 6.7 Distance on the Coordinate Plane Calculate the distance of two points on the coordinate plane C Given two points, A(-4, -3) and C(4, 4) Find the distance between A and C. A B 1. Draw point B(4, -3). 2. Find the distance of AB. 3. Find the distance of BC. 4. What kind of triangle is ABC. 5. What theorem can calculate AC? 6. AC = ( 3) 2 7. AC =

12 Chapter 6.7 Distance on the Coordinate Plane Calculate the distance of two points on the coordinate plane From this activity, we can write the following theorem: The Distance Formula Theorem: the distance between two points on the Cartesian plane with coordinates x 1, y 1 and x 2, y 2 is d = x 2 x y 2 y 1 1 Bookwork: page 266, problems 12-29

13 Discuss what is a Pythagorean Triple Pythagorean Triples The last section defined the Pythagorean Theorem as a 2 + b 2 = c 2. A Pythagorean Triple is a group of three whole numbers that satisfies the Pythagorean Theorem, where side c is the hypotenuse. There are 16 primitive Pythagorean triples with c 100: (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53) (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73) (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)

14 Discuss what is a Pythagorean Triple A B C X Y (0, 1) (0, -1) Perpendicular Pythagorean Triples

15 Discuss what is a Pythagorean Triple A B C X Y (0, 1) (0, -1) Perpendicular Pythagorean Triples

Geometry - Concepts 9-12 Congruent Triangles and Special Segments

Geometry - Concepts 9-12 Congruent Triangles and Special Segments Concept 9 Parallel Lines and Triangles (Section 3.5) ANGLE Classifications Acute: Obtuse: Right: SIDE Classifications Scalene: Isosceles: