Download Quadratics

Properties and
Attributes of Triangle
By: Ou Suk Kwon
• Perpendicular Bisector is a line that pass through a vertex of
one angle of triangle passing through the line that is in the
opposite side, this line must pass through the midpoint and
should be perpendicular.
• Perpendicular Bisector Theorem tells us if a point is on the
perpendicular bisector segment, then it is equidistant from
the endpoints of the segments.
• Its converse says if equidistant from the endpoints of a
segment, then it is one the perpendicular bisector of the
segment.
1) AD perpendicular BC
DB=DC
Conclusion: AB=AC
2) BF perpendicular AC
AF=CF
Conclusion: BA=BC
3) CE perpendicular AB
AE=BE
Conclusion: CA=CB
1) AB=AC
Conclusion: AD perpendicular BC
BD=CD
2) BA=BC
Conclusion: BF perpendicular AC
AF=CF
3) CA=CB
Conclusion: CE perpendicular AB
AE=BE
• Angle bisector is when you are bisecting each
angles of the triangle. You use them when you
are finding incenter of the triangle.
• Angle bisector theorem tells us if a point is on the
bisector of an angle, then it is equidistant from
the sides of the angle.
• And its converse tells us that 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.
1) BAD=CAD
Conclusion: BD=CD
XWD=YWD
Conclusion: XD=YD
XWD=YWD
Conclusion: XD=YD
1) XD=YD
Conclusion:
XWD= YWD
XD=YD
Conclusion:
XWD=YWD
2) XD=YD
Conclusion:
XWD=YWD
• Concurrent means that something is meeting,
intersecting or converging with something
else. The theorem is a point is on the
perpendicular segment iff it is equidistant to
all vertices. The circumcenter of a triangle is
always equidistant from the vertices of the
triangle.
• Circumcenter is the point where all the
perpendicular bisector segments meets.
• The angle bisectors of a triangle intersect at a
point that is equidistant from the sides of the
triangle. That is why you can make a circle in
side of that triangle and the radius will be
same.
• Incenter is the point that is intersected by all
three angle bisectors. It is equidistance from
each sides.
• Median is the segment that goes from the
vertex to the midpoint of the opposite side.
• A centroid is a point where all the medians
meet. Something very special about centroid
is that each line is divided into one side that is
one-third and the other side is two-thirds, also
centroid is the center of gravity when you are
building something.
Altitude is a perpendicular segment that
pass through the vertex and the side of the
opposite side to that vertex.
And simply orthocenter is the point where
every altitudes meet.
• Mid segment of the triangle is the segment
that has two endpoints of the mid points of 2
sides of the triangle.
• The segment joining the midpoints of two
sides of a triangle is parallel to the third side,
and is half the length of the third side.
• The Theorem tells us that the midsegment of
a triangle will be parallel to a side of triangle
and also will be ½ of it.
• The relationship between opposite angles and
segments is that the side that is the longest in
the triangle will have the biggest angle facing
it. The angle that is facing the shortest side
will be the smallest angle in the triangle.
• We use Angle-Side Relationships to find the
biggest and smallest angles just by knowing
the measurement of sides, or opposite way.
• An exterior angle is always bigger than the 2
non-adjacent angles in the inside of a triangle.
Examples
The sum of any 2 sides of the triangle will be
always greater then the 3rd side.
1)AB+AC > BC
AB+BC>AC
3) BA+VX>AC
2) BA+CA>BC
AC+BC>AC
Indirect proofs
1.Change the statement as the opposite(false)
2.Use the new statement as the given
3.Solve
4.When you can’t solve any more write a
contradiction.
• You just have to follow this steps to prove any
statement using indirect proofs.
Examples
• Triangle has 2 obtuse angles
• So first change the statement to opposite,
which is triangle has 2 obtuse angles (given).
• Then <1+<2+<3=180 (triangle) sum theorem
• Obtuse angle= 90>x (def of obtuse angle)
• When we have 2 angles that are obtuse we
will get more then 180 in all the triangle
(contradiction)
Example
• The women was in the bathroom, and at the
same time she was blamed that she Stoll
pancakes from monkeys that are in the jungle,
people saw her entering to bathroom.
• So here we change it to opposite she is guilty
because she steal monkey’s pancake in
jungle(given)
• She can’t be in 2 places at the same time
(contradiction)
Example
• Triangle can’t have angles that are more than
180
• Triangle has angles that are more than 180
(given)
• <1+<2+<3=180 (def of triangle)
• If one angle has more than 180, other 2 angles
won’t have here own angles, (contradiction)
Hinge theorem
• If 2sides of a triangle are congruent to other
two sides, then if one of the included angle of
the triangles is bigger, then the opposite side
is also bigger than the other.
• Its converse says, if 2sides of a triangle are
congruent to other two sides, then if one of
the third side of the triangles is bigger than
the other, then the opposite angle is also
bigger than the other.
Examples
Given: AC=DF CB=FE
Conclusion AB=DE
Given: AC=DF CB=FE
Conclusion AB=DE
Given: AC=DF CB=FE
Conclusion AB=DE
Converse examples
1<2
C<D
F<C
• Hypotenuse is length of leg times √2.
• Both legs are congruent.
• So something very special about this theorem
is that the triangle’s legs will be the same size
but the hypotenuse will be length times √2
Examples
15 X root2
X= 4 X root2
X= 4root2
16/root2
Which is 8root2
30-60-90 Theorem
• In the triangle that has 3 angles of 30
60 and 90 the Hypotenuse will be
always double of the base and the 3rd
side will be always base X root3
Examples
X= 16/2 = 8
Y= 8 X root3
8root3
X= 60/2 = 30
X=12/root3
4root3
y= 4root3 X 2
8root3
_____(0-10 pts.) Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse.
Give 3 examples of each.
_____(0-10 pts.) Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3
examples of each.
_____(0-10 pts.) Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle
theorem. Explain what a circumcenter is. Give at least 3 examples of each.
_____(0-10 pts.) Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least
3 examples of each.
_____(0-10 pts.) Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle
theorem. Give at least 3 examples of each.
_____(0-10 pts.) Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of
altitudes of a triangle theorem. Give at least 3 examples.
_____(0-10 pts.) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples.
_____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their opposite angles.
Give at least 3 examples.
_____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples.
_____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples.
_____(0-10 pts.) Describe how to write an indirect proof. Give at least 3 examples.
_____(0-10 pts.) Describe the hinge theorem and its converse. Give at least 3 examples.
_____(0-10 pts.) Describe the special relationships in the special right triangles (30-60-90 and 45-45-90). Give at least 3