1. Relationships Within Triangles
Chapter 5

2. 5.1 – Midsegment Theorem Midsegment of a triangle
Segment that connects midpoints of two sides of a triangle
Every triangle has three midsegments

3. Midsegment Theorem Theorem 5.1 – Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side
𝐷𝐸 || 𝐴𝐶 𝑎𝑛𝑑 𝐷𝐸= 1 2 𝐴𝐶
d E c B A

4. 5.2 – Use Perpendicular Bisectors
Segment, ray, line, or plane that is perpendicular to a segment at its midpoint
Equidistant
Points on the perpendicular bisector of a segment are the same distance from the segment’s endpoints

5. Theorems Theorem 5.2 – Perpendicular Bisector Theorem
In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
Theorem 5.3 – Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment

6. Example 1
Line BD is the perpendicular bisector of segment AC. Find the length of AC. C
B D A
3x + 14 5x

7. Concurrency of Lines, Rays & Segments
Concurrent
When three or more lines, rays, or segments intersect in the same point
Point of concurrency
Point of intersection of three or more lines, rays, or segments

8. Theorem 5.4 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
(If EG, EF, and ED are perp. Bisectors, then AG=BG=CG)

9. Example 3
Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor (D)
Find a location for the distributor that is equidistant from the three carts
(you will draw on the next slide)

10. Circumcenter Circumcenter Activity:
Point of concurrency of the three perpendicular bisectors of a triangle
Activity:
Construct the perpendicular bisectors of a triangle, and check to see if a circle can be constructed so that all vertices of the triangle lie on the circle

11. 5.3 – Use Angle Bisectors of a Triangle
Remember:
An angle bisector is a ray that divides an angle into two congruent adjacent angles
The distance from a point to a line is the length of the perpendicular segment from the point to the line

12. Theorems Theorem 5.5 – 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 5.6 – Converse of Angle Bisector Thrm.
If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle

13. Examples
Example 1
P. 312
Example 2
P. 313

14. Example 3 For what value of x does P lie on the bisector of <A?
GP #1-4 p. 313 B
x + 3 P A
2x – 1 C

15. Incenter of a Triangle
Theorem 5.7 – 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
Incenter
Point of concurrency of the three angle bisectors of a triangle
Always lies inside the triangle
Example 4, p. 314

16. 5.4 – Use Medians & Altitudes
Median of a triangle
Segment from a vertex to the midpoint of the opposite side
Centroid
Point of concurrency of the three medians of a triangle

17. Theorem Theorem 5.8 – Concurrency of Medians of 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 medians of PQR meet at point V
𝑃𝑉= 2 3 𝑃𝑇, 𝑄𝑉= 2 3 𝑄𝑈 𝑅𝑉= 2 3 𝑅𝑆

18. Altitudes of a Triangle
Altitude of a triangle
Perpendicular segment from a vertex to the opposite side of to the line that contains the opposite side
Theorem 5.9 – Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle are concurrent
Orthocenter
Point at which the lines containing the three altitudes of a triangle intersect

19. 5.5 – Inequalities in a Triangle
Theorem 5.10
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
Theorem 5.11
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

20. Triangle Inequality Theorem
Theorem 5.12 – Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side
Example 3
A triangle has one side of length 12 and another length 8. Describe the possible lengths of the third side.
GP #3 p.332