5.1 Perpendiculars and Bisectors
Perpendicular Bisector: a segment, ray, line, or plane that is perpendicular to a segment at its midpoint.
Equidistant from two points: a point whose distance from each point is the same.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
C
If CP is the perpendicular bisector of AB,
Then CA = CB
A P B
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 onthe A P B
Perpendicular bisector of AB.
D
Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
If m<BAD = m<CAD, then DB = DC
Converse of the Angle Bisector Theorem: 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.
If DB = DC, then m<BAD = m<CAD
Read pages 264 – 267
Do page 267: 1 – 13, 16 – 26, 33 - 35 all
5.2 Bisectors of a Triangle
Perpendicular bisector of a triangle: a line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side.
Concurrent lines: three or more lines, rays, or segments that intersect in the same point.
Point of concurrency: the point of intersection of the three lines.
Circumcenter of the triangle: the point of concurrency of the perpendicular bisectors of a triangle.
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. B
PA = PB = PC
A C
Incenter of a triangle: the point of concurrency of the angle bisectors of a triangle.
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. E
PD = PE = PF
D F
Read pages 272 – 274
Do page 275: 1 – 4, 10 – 21, 27 - 28 all
5.3 Medians and Altitudes
Perpendicular Angle Median Altitude
Bisector Bisector
Median: a segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side of the triangle.
Altitude: the perpendicular segment from a vertex to the opposite side.
Centroid: the point of concurrency of the medians of a triangle. The Centroid divides each median into a short and long segment. The long segment is twice the short and the short is half the long.
Centroid is 2/3 the distance from vertex to midpoint.
Orthocenter: the point of concurrency of the altitudes of a triangle.
The Orthocenter does nothing special.
Read pages 279 – 281. Yes – really read them
Do page 282: 1 – 19
5.4 Midsegment Theorem
Midsegment of a triangle: a segment that connects the midpoints of two sides of a triangle.
Midsegment Theorem: the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length.
C
DE ll AB
D E
DE = ½ AB
A B
How many midsegments are there in a single triangle?
Where are they located?
What is true about each midsegment?
AD = _____CE = _____
BF = _____DE = _____
DF = _____EF = _____
EF ll ____DF ll ____DE ll ____
Find the perimeter of each triangle:
ABC = _____ ADF = ____ DCE = ____ FEB = ____
Complete page 290: 1 – 18, 26 – 29, 32, 37
5.5 Inequalities in One Triangle
The longest side of a triangle is opposite the largest angle.
The shortest side of a triangle is opposite the smallest angle.
Write the measurements of the triangles in order from least to greatest.
Angles in order:Angles in order:
Sides in orderSides in order:
Exterior Angle Inequality: the measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. A
m<1 > m<A and m<1 > m<B
1
C B
Triangle Inequality Theorem: the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
AB + BC > AC AC + BC > AB AB + AC > BC
Find the possible side lengths of the following triangles:
One side is 10 inches and another is 8 inches
One side is 3 cm and another is 7 cm.
Can you have a triangle with sides of:
3 ft, 5 ft, and 9 ft. 5 in, 7 in, and 11 in 6 cm, 6 cm, and 12 cm
Read pages 295 – 297
Complete page 298: 1 – 31 all
5.6 Indirect Proof and Inequalities in Two Triangles
Indirect Proof: a proof in which you prove that a statement is true by first assuming that its opposite is true. If this assumption leads to an impossibility, then you have proved that the original statement is true.
Hinge Theorem: if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
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 triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
Examples: Complete with <, =, or >
AC ___ DF AC ___ DF <C ____ <F
Do page 305: 3 – 5, 7 – 20, 24 and Quiz # 2: page 308: 1 – 8
Chapter Test Five: page 313: 1 – 14
1. FG ll CE2. if FG = 8, then CE = 16
3. perimeter of GHE = 214. LQ, LM, MQ
5. QM, PM, QP6. MP, NP, MN
7. DE is longer than AB8. the second group
1. always2. sometimes
3. never4. sometimes
5. always6. HC = 12, HB = 10, HE = 5, BC = 19.8
7. centroid
8. altitude, angle bisector, median, and perpendicular bisector
9. EF = ½ AB, EF ll AB by the Midsegment Theorem
10. <ACB < <BAC because 16 < 19.8
11. Use angle bisectors to find the incenter which is equidistant from the sidewalks.
- Converse of the Hinge Theorem
- 10 feet – or just under 10 feet
- AC > BC