introducing

Chapter 5Relationships with Triangles

Chapter 5: Triangle Segments (MA.G.4.2 and MA.G.4.5)

• The 5 Segments we will discuss are:1. Midsegment (5.1)2. Perpendicular Bisector (5.2)3. Angle Bisector (5.3)4. Median (5.4)5. Altitude (5.4)6. After these segments are discussed we will then

move on to relationships between the sides and angles of a triangle (5.6 and 5.7)

5.1: Midsegments of Triangles

• Chapter 5 is all about 5 special “SEGMENTS” that can be drawn inside a triangle

• Review Quickly:– What’s a segment?– What does a segment have to have in order to be a

segment?• These segments have special properties and allow us to

find special points within the triangle that may be useful for various purposes.

• For example: one point can help you find the balancing point of the triangle, one point can help you find where to meet if 3 people are coming from different places

5.1: Midsegments of Triangles (MA.G.4.2 and MA.G.4.5)

A Midsegment is a segment that connect the midpoints of two sides of a triangle

Think: how many midsegments should I be able to draw in a triangle? If I draw all of them what have a created?

Draw a Triangle ABC , Find the midpoints of all 3 sides and label them D, E, and F. Draw all 3 Midsegments and Identify the following:

• Which segments on the perimeter of the triangle are equal? • What sides of the triangle are parallel to which

midsegments?

Special Properties of Midsegments: • Triangle Midsegment Theorem: A midsegment will be

half the length of the triangle side it is parallel too

5.2: Perpendicular and Angle Bisectors (MA.G.4.5)

A Perpendicular Bisector has two important characteristics:1) It bisects a side (cuts it in half or goes through the midpoint)2) It makes a 90 degree angle with the side

Think: how are midsegments similar to perpendicular bisectors and how are the two different?

Draw a Triangle ABC , Find the midpoints of all 3 sides and label them D, E, and F. Draw all 3 Perpendicular Bisector and Identify the following:

• Which segments on the perimeter of the triangle are equal?

Special Properties of Perpendicular Bisectors: • Perpendicular Bisector Theorem: If a point is on the “PB” then

it is equidistant from the endpoints of the segment. And Conversely, if a point is equidistant from the endpoints it must be on the perpendicular bisector.

CLASSWORK QUESTIONS

• Work in Groups to Solve the Following:–Page 288 #9-25 –Page 296 #16-23

EVEN ones will be graded next class for a HL Grade

5.3: Bisectors in Triangles

– http://www.khanacademy.org/math/geometry/triangles/v/circumcenter-of-a-triangle (8:00min)

– The Perpendicular Bisectors of the Triangle (We learned about them in 5.2) all meet at one point.

– Any time that lines meet, they intersect at a “POINT OF CONCURRENCY”

– The “point of concurrency” for perpendicular bisectors is called THE CIRCUMCENTER

– The CIRCUMCENTER has the special property that it is the same distance from each of the end points.

– The CIRCUMCENTER will be inside the triangle if the triangle is acute, on the hypotenuse if the triangle is right, and outside the triangle if the triangle is obtuse.

5.3: Bisectors in Triangles

http://www.khanacademy.org/math/geometry/triangles/v/circumcenter-of-a-triangle – The Angle Bisectors of a Triangle meet at a point

of concurrency called the “INCENTER”.– The Incenter has the special property that it is

equidistant from each side of the triangle. – It is also the center of a circle that has been

inscribed in a triangle.

CLASSWORK QUESTIONS

• Work in Groups to Solve the Following:–Page 305 # 7, 9, 15-18, 26, 28

5.4: Medians and Altitudes

– The median of a triangle runs from a vertex to the midpoint of the opposite side. The 3 medians will meet at a point of concurrency called the CENTROID. The centroid has the special property that is 2/3 of the way from the vertex the opposite side. It is also called the balancing point.

– The altitude of a triangle runs from a vertex to a 90 degree angle on the opposite side. This is also known as the height of a triangle. The altitudes will meet at a point called the ORTHOCENTER

5.6: Triangle Comparison and Inequality Theorems

–The longest side of a triangle is always opposite the largest angle.

–Two sides of a triangle must always add to be bigger than the 3rd in order for the triangle to exist.

Page 312 #8-13Page 329 # 9, 13, 17, 19, 21, 25,

CLASSWORK QUESTIONS

• Work in Groups to Solve the Following:–Page 312 # 8-13, 17, 19, 31