Chapter 5 – Properties of Triangles • The Bigger Picture • Properties of Triangles such as perpendicular and angle bisectors and how they relate in triangles • Congruency of perpendicular and angle bisectors • - The use of medians and altitudes to locate points on a triangle The “What” and the “Why” • Use properties of mid-segments of a triangle • - Determining length • Compare the length of sides or the measures of the angles of a triangle • Determining balance points for structural projects such as cranes and booms • Understand and write indirect proofs • Prove theorems that cannot be easily proved directly • Use the Hinge Theorem and its converse to compare side lengths and angle measures of triangles. • - Comparative analysis tool for measuring distance to a common point • Use of properties of perpendicular bisectors and angle bisectors • Optimize the positioning of a goalie in hockey or soccer • Use properties of perpendicular bisectors and angle bisectors • - Using the concept of equidistant to locate objects and points • Use properties of medians and altitudes of a triangle • - Find points on a triangle and use them to measure various objects such as a person’s heart fitness
A C B Properties of Triangles Soccer goalkeepers use triangle relationships to help block shots on goal. An opponent can shoot the ball from many different angles. The goalkeeper determines the best defensive position by imagining a triangle formed by the goal posts and the opponent. The opponent x is trying to score a goal. Which position do you think is best for the goalkeeper, A, B, or C? Why? Estimate the measure of <X, known as the shooting angle. How could the opponent change positions to improve the shooting angle?
C P C P A B A B 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. If CP is the _|_ bisector of AB, then CA = CB C P D Theorem 5.2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB A B Using Properties of Perpendicular Bisectors In chapter 1 we learned that a segment bisector intersects a segment at its midpoint. A segment, ray, line, or plane that is perpendicular to a segment at it’s midpoint is called a Perpendicular Bisector. CP is a _|_ bisector of AB
If we are told that MN is the perpendicular bisector of ST, how can we use that to help determine other aspects about the figure? What segment lengths in the diagram are equal? Why? Explain why Q is on MN. T 12 M N Q 12 S Using Perpendicular Bisectors
Q P m * P Using the Properties of Angle Bisectors Consider the following: the distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. And, when a point is the same distance from one line as it is from another, then the point is equidistant from the two lines. Knowing that, we can then apply the logic to a point on the interior of an angle, and use it to help determine the bisector of the angle.
B A D C B A D C Angle Bisector Theorems Theorem 5.3 Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is equidistant from the two sides of the angle. If m<BAD = m<CAD, then DB = DC Theorem 5.4 Converse of Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, the it lies on the bisector of the angle. If DB = DC, then m<BAD = m<CAD
Given: D is on the bisector of <BAC DB _|_ AB, DC _|_ AC Prove: DB = DC Plan: Prove /\ ADB = /\ ADC, Then Conclude DB = DC, so DB = BC B D A C ~ ~ Statements Reasons 1. <BAD = <CAD 1. Definition of an Angle Bisector 2. <ABD & <ACD are Right Angles 2. Definition of Perpendicular Line 3. <ABD = <ACD 3. Right Angle Congruence Theorem 4. AD = AD 4. Reflexive Property of Congruence 5. /\ ADB = /\ ADC 5. AAS Congruence Theorem 6. BD = DC 6. CPCTC 7. BD = DC 7. Definition of Congruent Segments ~ ~ ~ ~ ~ Proof of Theorem 5.3 – Angle Bisector Theorem
Acute Right Obtuse Bisectors of a Triangle Perpendicular Bisector of a Triangle: A line (ray of segment) that is perpendicular to the side of the triangle at it’s midpoint. Concurrent Lines: Three of more lines that intersect at the same point. Point of Concurrency: The point of intersection of the lines is called the point of concurrency. The three perpendicular bisectors of a triangle are concurrent. And the point of concurrency is known as the Circumcenter and can be located inside, on, or outside the triangle.
B P C A Theorem 5.5 Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. PA = PB = PC The point of concurrency (P) of the perpendicular bisectors of the triangle is called the Circumcenter of the Triangle Where and how might this be helpful in the real world?
Perpendicular Bisectors of a Triangle and Facilities Planning – an example Client F * Client E * * Client G A company is planning to build a new distribution facility that is convenient to all of it’s major clients. How might locating the circumcenter of the three clients be beneficial in determining the location?
Using Angle Bisectors of a Triangle An Angle Bisector of a triangle is a bisector of an angle of the triangle. The three angle bisectors are concurrent. The point of concurrency is called the incenter of the triangle. Theorem 5.6 – Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF
P R S L M Q N Using Angle Bisectors Based upon the Angle Bisector Theorem, which segments are congruent? If the length of ML = 17, and the length of MQ = 15, can we determine the length of LQ, LS, and LR?
A E F P B D C Medians and Altitudes of Triangles Median – a segment whose endpoints are a vertex and the midpoint of the opposite side of the triangle. The three Medians are Concurrent The point of concurrency is called the Centroid of the triangle 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. If P is the centroid of /\ ABC, then, AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3CE
Acute Right Obtuse Altitude of a Triangle An altitude of a triangle is a perpendicular segment from a vertex to the opposite side of a triangle, or to a line that contains the opposite side of the triangle. An altitude can lie inside, outside, or on the triangle. Altitudes are concurrent, and the point of concurrency is called the orthocenter of the triangle.
Acute Right Obtuse Summary – Triangle Bisectors Perpendicular Bisector of a Triangle: A line (ray of segment) that is perpendicular to the side of the triangle at it’s midpoint. The point of concurrency is called the Circumcenter. An Angle Bisector of a triangle is a bisector of an angle of the triangle. The point of .concurrency is called the Incenter. An altitude of a triangle is a perpendicular segment from a vertex to the opposite side of a triangle, or to a line that contains the opposite side of the triangle. An altitude can lie inside, outside, or on the triangle. Altitudes are concurrent, and the point of concurrency is called the orthocenter of the triangle. Median – a segment whose endpoints are a vertex and the midpoint of the opposite side of the triangle. The three Medians are Concurrent and the point of concurrency is called the Centroid of the triangle
C D E A B Mid-Segment Theorem A Mid-segment of a triangle is a segment that connects the midpoints of two sides of a triangle. Theorem 5.9 Mid-segment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side, and is half as long. DE || AB and DE = ½ AB
Proving the Mid-segment Theorem 1. DE is the mid-segment for sides AC and BC. 2. Using the midpoint formula, we can determine the coordinate values for points D and E. D = (5, 5) E = (11, 5) 3. Determine the slope of DE - Compare that to the slope of AB 4. Determine the length of DE and AB using the distance formula. - Since they are both horizontal lines, the length can be determined as the absolute value of the difference in the x values. DE = 6 AB = 12 5. DE || AB & DE = ½ AB C(10, 10) D E B (12, 0) A(0, 0)
Using Midpoints to Draw a Triangle The midpoints of the sides of a triangle are L (4, 2), M(2, 3), and N(5, 4). What are the coordinates of the vertices of the triangles?
Hinge Theorem Theorem 5.14 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second. Theorem 5.15 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. Indirect Proof of Theorem 5.15 – Example 2
__ __ __ __ ~ ~ • If AB = DE and BC = EF, AC = 12, m<B = 36*, m<E = 80*. Which of the following is a possible length for DF? 8, 10, 12, or 23? • In a triangle RST and triangle XYZ, RT = XZ, ST = YZ, RS = 3.7, XY = 4.5, and m<Z = 75*. Which of the following is a possible measure for <T? 60*, 75*, 90*, or 105*? __ __ __ __ __ ~ ~ Finding Possible Side Lengths and Angle Measures E B 80 36 D F A C
Comparing Distances using the Hinge Theorem You and a friend are flying separate planes. You leave the airport and fly 120 miles due west. You then change direction and fly W 30* N for 70 miles. Your friend leaves the airport and flies 120 miles due east. She then changes direction and flies E40* S for 70 miles. Each of you have flown exactly 190 miles, but which one of you is farther from the airport? P airport
Comparing Distances using the Hinge Theorem • Your flight: 100 miles due west, then 50 miles N20* W. • Your Friend: 100 Miles due north, then 50 miles N30* E • 2. Your flight: 210 miles due south, then 80 miles S70* W. • Your Friend: 80 miles due north, then 210 miles N50* E. P airport