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MELANIE DOUGHERTY
GEOMETRY JOURNAL 5
Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse.
A perpendicular bisector is a line perpendicular to the base of a triangle that bisects it.
Perpendicular Bisector theorem: If a point is on the perpendicular bisector of a
segment, then it is equidistant form the endpoints of the segment.
Converse: if a point is equidistant from the endpoints of a
segment, then it is on the perpendicular bisector of the segment.
Perpendicular Bisector Examples
AB = AC
AB = AC
AC = BC
PB Converse Examples
LN = EN
AD = DC
CD = DB
Describe what an angle bisector is. Explain the angle bisector theorem and its converse.
An angle bisector is a line that divides the angle.
The angle Bisector theorem: If a point is on the bisector of an angle, then it is
equidistant from the sides of the angleConverse:
If a point is equidistant from the sides of an angle the it is on the bisector.
Angle Bisector theorem examples
BF = FC
<UFK is congruent to <KFC
<EWR is congruent to <RWT
CONCURRENT
When 3 or more lines intersect at one point
Concurrency of perpendicular bisector theorem of triangles
The circumcenter of a triangle is equidistant from the vertices of the triangle.
Circumcenter: where the 3 perpendicular bisectors of a triangle meet
circumcentercircumcent
er
circumcenter
DA = DB = DC
DA = DB = DC
DA = DB = DC
acute
right
obtuse
concurrency of angle bisectors of a triangle theorem
Incenter of a triangle : where the 3 angle bisectors of a triangle meet
Concurrency of a angle bisectors of a triangle theorem: the incenter of a triangle is equidistant from the sides of the triangle.
incenter
incenter
incenter
DF = DG = DE
DF = DG = DE
DF = DG = DE
ACUTERIGHT
OBTUSE
MEDIANS AND ALTITUDES OF TRIANGLES
The median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side
The centroid of a triangle is the point of concurrency of the medians of a triangle.
Concurrency of medians of a triangle theorem: the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.
EXAMPLES
MEDIAN CENTROID
CMTT
Concurrency of altitudes of triangles theorem
Altitude: a perpendicular segment from a vertex to the line containing the opposite side
Orthocenter: point where the 3 altitudes of a triangle meet.
Concurrency of altitudes of triangles theorem: the lines containing the altitude are concurrent
Triangle Midsegment theorem
A midsegment is a segment that joins the midpoints of two sides of a triangle
Midsegment theorem: a midsegment of a triangle is parallel to a side of the triangle, and its length is half of that side.
midsegment midsegme
ntmidsegment
AB ll EF, EF = ½ AB
DE ll BC, DE = ½ BC
DE ll AC, DE = ½ AC
Angle-Side Relationship in Triangles
If none of the sides of the triangle are congruent then the largest side is opposite the largest angle.
If none of the sides of the triangle are congruent then the shortest side is opposite the smallest angle.
EXAMPLES
Triangle Inequality
The sum of the lengths of two sides of a triangle is greater than the length of the third side.
Writing an indirect proof
1. Identify what is being proven2. Assume that the opposite of your conclusion
is true3. Use direct reasoning to prove that the
assumption has a contradiction4. Assume that if the 1st assumption is false
then what is being proved is true.
EXAMPLES
Step 1Given: triangle JKL is a right triangleProve: triangle JKL doesn't have and obtuse angleStep 2 Assume <K is an obtuse angleStep 3m<K + m<L = 90 m<K = 90 – m<L m<K > 90 90 – m<L > 90 m<L <0 (this is impossible)Step 4 The original conjecture is true.
Hinge theorem
If 2 sides of a triangle are congruent to 2 sides of an other triangle and included angles are not congruent, then the longer third side is across from the larger included angle.
Converse: if 2 sides of 2 triangle are congruent to 2 sides of an other triangle and the third sides of an other triangle are not congruent, then the larger included angle is across from the longer third side.
Examples