Properties of Triangles Chapter 5 Properties of Triangles

Chapter 5 Objectives Identify a perpendicular bisector Identify characteristics of angle bisectors Visualize concurrency points of a triangle Compare measurements of a triangle Display the midsegment of a triangle Utilize the triangle inequality theorem Create an indirect proof

Perpendiculars and Bisectors Lesson 5.1 Perpendiculars and Bisectors

Lesson 5.1 Objectives Define perpendicular bisector Utilize the Perpendicular Bisector Theorem and its converse

Perpendicular Bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

Equidistant In order for an object to be equidistant from two or more locations, the following must be true: The distance to each object must be equal. The segment drawn from the object must intersect each location at the same angle.

Theorem 5.1: Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Theorem 5.2: Perpendicular Bisector Converse If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Example 1 Tell whether there is enough information that C lies on the perpendicular bisector of segment AB. Explain. Yup! Yup! C is equidistant from A and B C is equidistant from A and B

Theorem 5.3: Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Theorem 5.4: Angle Bisector Converse If a point is on the interior of an angle and it is equidistant from the two sides of the angle, then it lies on the bisector of an angle.

Example 2 Can you conclude that ray BD bisects ABC? Explain. Yup! Nope! We do not know the angles at which the segments intersect the sides of ABC. D is equidistant from A and C

Lesson 5.1 Homework In Class HW Due Tomorrow 1-7 8-13, 16-26, 41-52 p268-271 HW 8-13, 16-26, 41-52 Due Tomorrow

Bisectors of a Triangle Lesson 5.2 Bisectors of a Triangle

Lesson 5.2 Objectives Define concurrency Identify the concurrent points inside triangles. Identify perpendicular and angle bisectors in a triangle. Differentiate between circumcenter and incenter

Perpendicular Bisectors of a Triangle A perpendicular bisector of a triangle is any segment or ray or line that is perpendicular to the midpoint of any side of a triangle.

Concurrency Concurrent lines exist when 3 or more lines, segments, or rays intersect at a common point. The point at which the concurrent lines intersect is called the point of concurrency.

Theorem 5.5: Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle will intersect to form a point of concurrency equidistant from the vertices. Hint: If the segment is perpendicular to a side, then it is equidistant to the vertices.

Circumcenter The point of concurrency of perpendicular bisectors in a triangle is called the circumcenter of a triangle. It is called this because it forms the center of a circle that is drawn connecting the vertices of the triangle. Notice the vertices of the triangle lie on the circumference of the circle. Thus the name circum-center.

Example 3 Find the following quantities: MO PR MN SP MP 26.8 26 40 22 44

Inside Out Type of Triangle Picture Point of Concurrency Inside Acute Triangle Right Triangle Obtuse Triangle Picture Point of Concurrency Inside On One Side Outside Note: All lines drawn must be perpendicular bisectors of the triangle sides.

Theorem 5.6: Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. Hint: When angles are equal, then the distance to the side is equal. Hint: But the perpendicular segments are not bisectors.

Incenter The point of concurrency of the angle bisectors is called the incenter of the triangle. It is called this because it creates the center of a circle formed by touching each side of the triangle once. Notice the circle formed is inside the triangle. Thus the name in-center.

Example 4 Point T is the incenter of PQR. Find ST. 15

Example 5 Three snack carts sell frozen yogurt at locations A, B, and C. The distributor for the snack carts wants to build a warehouse that is equal distance to all three carts. Describe how the distributor could find a location for the warehouse. Show where the warehouse should be built on the map. Mr. Lent’s Ice Cream Warehouse Use the perpendicular bisectors of a triangle to determine the circumcenter of the three locations. The circumcenter is equidistant from all vertices of a triangle.

Lesson 5.2 Homework In Class 1-4 p275-278 HW 5-21, 24-28 Due Tomorrow

Medians and Altitudes of Triangles Lesson 5.3 Medians and Altitudes of Triangles

Lesson 5.3 Objectives Define a median of a triangle Identify a centroid of a triangle Define the altitude of a triangle Identify the orthocenter of a triangle

Triangle Medians A median of a triangle is a segment that does the following: Contains one endpoint at a vertex of the triangle, and Contains the other endpoint at the midpoint of the opposite side of the triangle. A B C D

Remember: All medians intersect the midpoint of the opposite side. Centroid When all three medians are drawn in, they intersect to form the centroid of a triangle. This special point of concurrency is the balance point for any evenly distributed triangle. In Physics, we would call it the center of mass. Obtuse Acute Right

Theorem 5.7: Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. The centroid is 2/3 the distance from any vertex to the opposite side. AP = 2/3AE 2/3BF BP = 2/3BF CP = 2/3CD 2/3AE 2/3CD

Example 6 S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following: RV 6 RU 4 is 2/3 of 6 Divide 4 by 2 and then muliply by 3. Works everytime!! SU 2 RW 12 TS 6 is 2/3 of 9 SV 3

Altitudes An altitude of a triangle is the perpendicular segment from a vertex to the opposite side. It does not bisect the angle. It does not bisect the side. The altitude is often thought of as the height. While true, there are 3 altitudes in every triangle but only 1 height!

Orthocenter The three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle. The orthocenter can be located: inside the triangle outside the triangle, or on one side of the triangle Obtuse Right Acute The orthocenter of a right triangle will always be located at the vertex that forms the right angle.

Theorem 5.8: Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent.

Example 7 Is segment BD a median, altitude, or perpendicular bisector of ABC? Hint: It could be more than one! Perpendicular Bisector Altitude None Median

Lesson 5.3 Homework In Class HW Due Tomorrow Quiz Tuesday 1-7 p282-284 HW 8-23, 39-45 Due Tomorrow Quiz Tuesday November 20

Midsegment Theorem of Triangles Lesson 5.4 Midsegment Theorem of Triangles

Lesson 5.4 Objectives Create the midsegment of a triangle Identify the characteristics of a midsegment of a triangle

Midsegment of a Triangle So far we have studied 4 types of special segments of triangles. Perpendicular Bisector Angle Bisector Median Altitude It just so happens that all of these intersect only one side at a time. And three of the four intersect an vertex and a side. Another type of special segment is one that connects the midpoints of the sides of a triangle. This special segment is called the midsegment of a triangle. Notice there are 3 midsegments in every triangle.

Theorem 5.9: Midsegment Theorem of a Triangle The segment connecting the midpoints of the two sides of a triangle is: Parallel to the third side Half the length of the third side The side it is parallel to DE = 1/2AC Segment DE // Segment AC

Example 8 Segment MP is the midsegment of LNO. Find x MP = ½NO P is the midpoint x = ½(16) 7 = ½(x) x = 4 x = 8 x = 14

Example 9 Fill in the following Segment GJ is parallel to ________. segment DF Segment EJ is congruent to _________. segment JF Segment DE is parallel to __________. segment KJ If EF = 18, then GK = _____. 9 If JK = 13, then ED = _____. 26

Lesson 5.4 Homework In Class HW Due Tomorrow 1-11 p290-293 HW 12-18, 21-29, 39-49 odds Due Tomorrow

Inequalities in One Triangle Lesson 5.5 Inequalities in One Triangle

Lesson 5.5 Objectives Compare angle sizes based on side lengths Utilize the Triangle Inequality Theorem

Theorem 5.10: Side Lengths of a Triangle Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Basically, the larger the side, the larger the angle opposite that side. 2nd Largest Angle Longest side Smallest Side Smallest Angle Largest Angle 2nd Longest Side

Theorem 5.11: Angle Measures of a Triangle Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Basically, the larger the angle, the larger the side opposite that angle. 2nd Largest Angle Longest side Smallest Side Smallest Angle Largest Angle 2nd Longest Side

Example 10 Name the smallest and largest angle. Largest Smallest

Example 11 Name the smallest and largest side. Largest Smallest

Theorem 5.13: Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 3 4 4 3 2 1 6 6 6 Add each combination of two sides to make sure that they are longer than the third remaining side.

Example 12 Determine whether the following could be lengths of a triangle. 6, 10, 15 6 + 10 > 15 10 + 15 > 6 6 + 15 > 10 YES! 11, 16, 32 11 + 16 < 32 NO! Hint: A shortcut is to make sure that the sum of the two smallest sides is bigger than the third side. The other sums will always work.

Homework 5.5 In Class 1-5 p298-301 HW 6-30 evens Due End of the Hour