Properties of Triangles
Vocabulary Words
1. Equidistant2. Locus3. Concurrent4. Point of concurrency5. Circumcenter6. Median7. Centroid8. Altitude9. Orthocenter10. Incenter
Perpendicular and Angle Bisectors
Equidistant – when a point is the same distance from two o r more objects.
Theorems:Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
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.
Example
Find each measure.
A. PB B. AB
C. AD
Distance and Angle Bisectors
Locus – a set of points that satisfies a given condition exp: The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.
Angle Bisector Theorem – If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem – If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
Example
Find each measure
A. ED
B. given that = 112
C. ,
Bisectors of Triangles
Concurrent – when three or more lines intersect at one point.
Point of concurrency – the point where three or more lines intersect.
Circumcenter of the triangle – the point of concurrency of the three perpendicular bisectors of a triangle.
Cirmcumcenter Theorem
The circumcenter of a triangle is equidstant from the vertices of the triangle.
Example
, , and are perpendicular bisectors of . Find HZ.
Your turn1. GM2. GK3. JZ
Incenter Theorem
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle.
Incenter Theorem – The incenter of a triangle is equidistant from the sides to the triangle.
Unlike the circumcenter, the incenter is always inside the triangle.
Example
JV and KV are angle bisectors of
A. The from V to KL.
Your Turn
QX and RX are angle bisectors of Find each measure.
1. The distance from X to PQ 19.22. m 52
Name Type Point of Concurrency
Perpendicular Bisector A line segment with the midpoint of a side as an end point
Circumcenter
Angle Bisector Bisects an angle on the interior of the triangle into two congruent angles
Incenter
Median A line segment with endpoints from a vertex and the midpoint of the opposite side
Centroid
Altitude Is a line segment from a vertex that is perpendicular to the side opposite the vertex
Orthocenter
Inequalities and Triangles
Page 247
Resource Book page 17
Review Exterior Angle TheoremPage 248 Exterior Angle Inequality Theorem
One Triangle Inequality
Page 261
Resource Book page 29
Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle.
Determine whether the measures and
can be lengths of the sides of a triangle.
Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle.
Check each inequality.
Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.
In and Find the range of the third side.
Inequalities Involving Two Triangles
Resource Book page 35