Slide 1: More About Triangles § 6.1 Medians § 6.4 Isosceles Triangles § 6.3 Angle Bisectors of Triangles § 6.2 Altitudes and Perpendicular Bisectors § 6.6 The Pythagorean Theorem § 6.5 Right Triangles § 6.7 Distance on the Coordinate Plane
Slide 2: Medians You will learn to identify and construct medians in triangles What You'll Learn 1) ______ 2) _______ 3) _________ Vocabulary median centroid concurrent
Slide 3: Medians In a triangle, a median is a segment that joins a ______ of the triangle and the ________ of the side __________________. vertex midpoint opposite that vertex C B A D E F centroid The medians of Δ ABC, AD, BE, and CF, intersect at a common point called the ________. When three or more lines or segments meet at the same point, the lines are __________. concurrent
Slide 4: Medians There is a special relationship between the length of the segment from the vertex to the centroid D C B A E F and the length of the segment from the centroid to the midpoint .
Slide 5: Medians Theorem 6 - 1 The length of the segment from the vertex to the centroid is _____ the length of the segment from the centroid to the midpoint. twice x 2x When three or more lines or segments meet at the same point, the lines are __________. concurrent
Slide 6: Medians D C B A E F CD = 14
Slide 7: End of Section 6.1
Slide 8: Altitudes and Perpendicular Bisectors You will learn to identify and construct _______ and __________________ in triangles. What You'll Learn 1) ______ 2) __________________ Vocabulary altitudes perpendicular bisectors altitude perpendicular bisector
Slide 9: Altitudes and Perpendicular Bisectors In geometry, an altitude of a triangle is a ____________ segment with one endpoint at a ______ and the other endpoint on the side _______ that vertex. perpendicular vertex opposite D The altitude AD is perpendicular to side BC. C A B
Slide 10: Altitudes and Perpendicular Bisectors C A B Constructing an altitude of a triangle 1) Draw a triangle like Δ ABC 2) Place the compass point on B and draw an arc that intersects side AC in two points. Label the points of intersection D and E. 3) Place the compass point at D and draw an arc below AC. Using the same compass setting, place the compass point on E and draw an arc to intersect the one drawn. 4) Use a straightedge to align the vertex B and the point where the two arcs intersect. Draw a segment from the vertex B to side AC. Label the point of intersection F. D E
Slide 11: Altitudes and Perpendicular Bisectors An altitude of a triangle may not always lie inside the triangle. Altitudes of Triangles acute triangle right triangle obtuse triangle The altitude is _____ the triangle The altitude is _____ of the triangle The altitude is _______ the triangle inside out side a side
Slide 12: Altitudes and Perpendicular Bisectors Another special line in a triangle is a perpendicular bisector . A perpendicular line or segment that bisects a ____ of a triangle is called the perpendicular bisector of that side. side A B C D D is the midpoint of BC. m altitude Line m is the perpendicular bisector of side BC.
Slide 13: Altitudes and Perpendicular Bisectors In some triangles, the perpendicular bisector and the altitude are the same. X Z Y E YE is an altitude. The line containing YE is the perpendicular bisector of XZ. E is the midpoint of XZ.
Slide 14: End of Section 6.2
Slide 15: Angle Bisectors of Triangles You will learn to identify and use ____________ in triangles. What You'll Learn 1) ___________ Vocabulary angle bisectors angle bisector
Slide 16: Angle Bisectors of Triangles Recall that the bisector of an angle is a ray that separates the angle into two congruent angles. S Q R P
Slide 17: Angle Bisectors of Triangles An angle bisector of a triangle is a segment that separates an angle of the triangle into two congruent angles. One of the endpoints of an angle bisector is a ______ of the triangle, and the other endpoint is on the side ________ that vertex. vertex opposite A B D C Just as every triangle has three medians , three altitudes , and three perpendicular bisectors , every triangle has three angle bisectors .
Slide 18: Angle Bisectors of Triangles Special Segments in Triangles Segment Type Property altitude perpendicular bisector angle bisector line segment line ray line segment line segment from the vertex, a line perpendicular to the opposite side bisects the side of the triangle bisects the angle of the triangle
Slide 19: End of Section 6.3
Slide 20: Isosceles Triangles You will learn to identify and use properties of _______ triangles. What You'll Learn 1) _____________ 2) ____ 3) ____ Vocabulary isosceles isosceles triangle base legs
Slide 21: Isosceles Triangles Recall from §5-1 that an isosceles triangle has at least two congruent sides. The congruent sides are called ____. legs The side opposite the vertex angle is called the ____. base In an isosceles triangle, there are two base angles, the vertices where the base intersects the congruent sides. vertex angle leg leg base base angle base angle
Slide 22: Isosceles Triangles Theorem 6-2 Isosceles Triangle Theorem 6-3 If two sides of a triangle are congruent, then the angles opposite those sides are congruent. 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. A B C A B C D
Slide 23: Isosceles Triangles Theorem 6-4 Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. A B C Theorem 6-5 A triangle is equilateral if and only if it is equiangular.
Slide 24: End of Section 6.4
Slide 25: Right Triangles You will learn to use tests for _________ of ____ triangles. What You'll Learn 1) _________ 2) ____ Vocabulary congruence right hypotenuse legs
Slide 26: Right Triangles In a right triangle, the side opposite the right angle is called the _________. hypotenuse hypotenuse The two sides that form the right angle are called the ____. legs leg leg
Slide 27: Right Triangles Recall from Chapter 5, we studied various ways to prove triangles to be congruent: In §5-5 , we studied two theorems SSS and SAS A B C R S T A B C R S T
Slide 28: Right Triangles Recall from Chapter 5, we studied various ways to prove triangles to be congruent: In §5-6 , we studied two theorems ASA and AAS R S T A B C R S T A B C
Slide 29: Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-6 LL Theorem If two legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. A C B D F E SAS same as
Slide 30: Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-7 HA Theorem If ______________ and an ( either ) __________ of one right triangle are congruent to the __________ and _________________ of another right angle, then the triangles are congruent. AAS same as the hypotenuse acute angle hypotenuse corresponding angle A C B D F E
Slide 31: Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-6 LA Theorem If one (either) ___ and an __________ of a right triangle are congruent to the ________________________ of another right triangle, then the triangles are congruent. ASA same as leg acute angle corresponding leg and angle A C B D F E
Slide 32: Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Postulate 6-1 HL Postulate If the hypotenuse and a leg on one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. A C B D F E ASS Theorem?
Slide 33: End of Section 6.5
Slide 34: Pythagorean Theorem You will learn to use the __________ Theorem and its converse. What You'll Learn 1) _________________ 2) _______________ * 3) _______ Vocabulary Pythagorean Pythagorean Theorem Pythagorean triple converse
Slide 35: Pythagorean Theorem If ___ measures of the sides of a _____ triangle are known, the ___________________ can be used to find the measure of the third ____. two right Pythagorean Theorem side a b c A _________________ is a group of three whole numbers that satisfies the equation c 2 = a 2 + b 2 , where c is the measure of the hypotenuse. Pythagorean triple 3 5 4 5 2 = 3 2 + 4 2 25 = 9 + 16
Slide 36: Pythagorean Theorem Theorem 6-9 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse __ , is equal to the sum of the squares of the lengths of the legs __ and __ . c b a c b a Theorem 6-10 Converse of the Pythagorean Theorem If c is the measure of the longest side of a triangle, a and b are the lengths of the other two sides, and c 2 = a 2 + b 2 , then the triangle is a right angle.
Slide 37: End of Section 6.6
Slide 38: Distance on the Coordinate Plane You will learn to find the ______________________on the coordinate plane. What You'll Learn Nothing new! You learned this in Algebra I. Vocabulary distance between two points
Slide 39: Distance on the Coordinate Plane 1) On grid paper, graph A(-3, 1) and C(2, 3). Hands-On! y x 2) Draw a horizontal segment from A and a vertical segment from C. A(-3, 1) C(2, 3) B(2, 1) 3) Label the intersection B and find the coordinates of B. QUESTIONS: What is the measure of the distance between A and B ? What is the measure of the distance between B and C ? What kind of triangle is Δ ABC ? If AB and BC are known, what theorem can be used to find AC ? What is the measure of AC ? (x 2 – x 1 ) = 5 (y 2 – y 1 ) = 2 right triangle Pythagorean Theorem 29 ≈ 5.4
Slide 40: Distance on the Coordinate Plane Theorem 6-11 Distance Formula If d is the measure of the distance between two points with coordinates (x 1 , y 1 ) and (x 2 , y 2 ), y x A(x 1 , y 1 ) B(x 2 , y 2 ) d then d =
Slide 41: Find the distance between each pair of points. Round to the nearest tenth, if necessary. 5 4.5
Slide 42: End of Section 6.7
Slide 43: Distance on the Coordinate Plane