5.1 Midsegments of Triangles

Midsegment- A segment connecting the midpoints of 2 sides of a triangle.

Find the value of x.

3x = 12(54 ) 2x =

12(10 x−21)

3x = 27 4x = 10x - 21

x = 9 -6x = -21

x = 3.5

5.2 Perpendicular and Angle Bisectors

Find QR.

3n -1 = 5n -7

6 = 2n

n = 3

QR = 8

5.3 Bisectors in Triangles

Point of concurrency- They point where 3 or more lines intersect

Circumcenter- The point of concurrency of perpendicular bisectors.

In the diagram, the perpendicular bisectors of triangle MNP meet at point O and are shown dashed. Find the indicated measure.

(6,5)

Incenter- the point of concurrency of angle bisectors

C Z

Find x.

X = 2 x = 4

5.4 Medians and Altitudes

Median- the segment from a vertex to the midpoint on the opposite side

Centroid- the point of concurrency of medians

Altitude- Perpendicular segment from a vertex to the opposite side.

Orthocenter – The point of concurrency for altitudes

5.6 Inequalities in One Triangle

The longest side and the largest angle are opposite each other.

The shortest side and smallest angle are opposite each other.

If we take three line segments, we may or may not be able to form a triangle with them.

For these three attempted triangle constructions, only the first set of side lengths forms a triangle.

Ex: Can the triangle have sides with the given lengths?

1.) 2, 3, 6 2.) 11, 15, 12 3.) 9, 5, 4

No yes no

Finding possible side lengths:

Ex: A triangle has one side of length 14 and another of length 10. Describe the possible lengths of the third side.

Assume x is the smallest side Assume x is the largest side

Ex: If 2 sides are 4 and 7, what are the possible lengths for the 3rd side?

x + 4 > 7 4 + 7 > x

x > 3 11 > x

3 < x < 11

5.7 Inequalities in Two Triangles

NL > QO