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Journal Chapter
5 Kirsten Erichsen
• Perpendicular Bisector and Theorem
•Angle Bisector and Theorem• Concurrency • Concurrency of
Perpendicular Bisectors• Circumcenter
• Concurrency of Angle Bisectors• Incenter• Median• Centroid
• Concurrency of Medians• Altitude of a Triangle
• Orthocenter• Concurrency of Altitudes• Midsegment and Theorem• Longer and Shorter Sides•Angle and Triangle inequality
• Indirect Proofs• Hinge Theorem
• Triangle Relationships
9-5
What is a Perpendicular Bisector?
A perpendicular bisector is a line that bisects a segment and is perpendicular to the line being bisected.
The perpendicular bisector passes right through the middle of the line or segment.
Example 1.
The Perpendicular Bisector, bisects the segment or line right through the middle.
The perpendicular bisector bisects right through the middle of the segment.
Example 2.
The perpendicular bisector bisects right through the middle of the segment.
The Perpendicular Bisector passes through the line to make up four 90° angles.
Example 3.
The perpendicular bisector bisects right through the middle of the segment.
Remember, the perpendicular bisector bisects through the middle, so both sides of the line after they have been bisected are congruent.
Perpendicular Bisector Theorem.
• If a point is on the perpendicular bisector, then it is equidistant to both endpoints of the segment.• CONVERSE: If a point is
equidistant to both endpoints, then it lies on the perpendicular bisector.
Example 1.
The measurements from point P to point J are also 6 cm.
Since point P lies on the Perpendicular Bisector, then it is has the same measure to both sides. So in conclusion, the measurement from P to J is 6 centimeters.
Example 2.
The measurement from point G to point C is 9 cm.
Since point P lies on the Perpendicular Bisector, then it is has the same measure to both sides. So in conclusion, the measurement from G to C is 9 centimeters.
Example 3.
The measurement from point J to point L is 15 cm.
Since point P lies on the Perpendicular Bisector, then it is has the same measure to both sides. So in conclusion, the measurement from J to L is 15 centimeters.
What is an Angle Bisector?
An Angle Bisector is a ray or line that cuts an angle into 2 congruent angles.
The angle bisector that cuts the angle, always has to be in the interior or inside of the angle.
Example 1.
The ray (purple) is the angle bisector.
The right angle is bisected into 2 congruent angles, both measuring 45°.
Example 2.
The ray (green) is the angle bisector.
The obtuse angle measures 140°. Since it has been bisected, both angles measure 70°.
Example 3.
The ray (green) bisects the acute angle.
The acute angle measures 45°. Since it has been bisected by a ray, then each one of the angles measures 22.5°.
Angle Bisector Theorem.
• Any point that lies on the angle bisector, is equidistant to both sides of the angle. • CONVERSE: If it is equidistant
to both sides of the angle, then it lies on the angle bisector.
Example 1.
The length from point D to point E is 15 centimeters.
To find the length of the other length, then follow the Angle Bisector Theorem. If it lies on the bisector then it is congruent to both sides.
Example 2.
The length from point J to N is 9 centimeters.
Both sides are congruent if the point lies in the Angle Bisector. The length from J to K is congruent to the length of J to N.
Example 3.
The length from point W to point X is 35 centimeters.
Both sides are congruent if the point lies in the Angle Bisector. The length from W to V is congruent to the length of W to X.
What is Concurrency?
Concurrent: it is said when three or more lines intersect at one point.
The point of intersection is called the point of concurrency.
Example 1.
The point of Concurrency is point A.
All 3 lines are concurrent because they intersect at the same point of concurrency.
A
Example 2.
The point of Concurrency is point J.
All 4 lines are concurrent because they intersect at the same point of concurrency.
J
Example 3.
The point of Concurrency is point L.
All 4 lines are concurrent because they intersect at the same point of concurrency.
L
Concurrency of Perpendicular Bisectors.
• The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle.
• This is also called the circumcenter.
Example 1.
Example 2.
Example 3.
It is 6 cm to the endpoints, because it lies on the Perpendicular Bisector
What is the Circumcenter?
• The circumcenter is the point of concurrency of the three bisectors of a triangle.
• The circumcenter of a triangle is equidistant from the vertices of the triangle.
Example 1.
The Perpendicular Bisectors starts from the Midpoint and make a straight line.
In a right triangle the circumcenter is always meet and are going to be on the lines or segments of the triangle.
Example 2.
The lines always start from the midpoint and make a straight line.
In an obtuse triangle the circumcenter is always outside of the triangle no matter what.
Example 3.
You always start from the midpoint of each side and make a straight line (PB).
In an acute triangle the circumcenter is always in the inside of the triangle no matter what.
Concurrency of Angle Bisectors.
• The angle bisectors are also concurrent to each other.
• Since a triangle has three angles, it has three angle bisectors that are concurrent.
• It can also be referred as the incenter.
Example 1.
All of the lines in blue are concurrent because they intersect at the same place. The lines in red are congruent because they have the same
lengths.
Example 2.
The lines in red are the same lengths from the point of concurrency to the sides of the triangle.
Example 3.
The lines in red are the same lengths from the point of concurrency to the sides of the triangle.
What is the Incenter?
• INCENTER: it is the point of concurrency of the three angle bisectors of a triangle.
• The incenter of a triangle is equidistant from the sides of the triangle.
• It is always going to be inside the triangle.
Example 1.
The incenter in an acute triangle is always going to be inside.
Example 2.
The incenter in a right triangle is always going to be inside.
Example 3.
The incenter in an obtuse triangle is always going to be inside.
What is the Median?
• MEDIAN: a segment whose end points are a vertex of a triangle and a midpoint.
• The midpoint has to be from the opposite side of the triangle.
C
D
Example 1.
The Median of this triangle is from point K to point J, but there can be 2 more.
J
K
I
H
Example 2.
The median being shown is from point L to point M.
M
L
N O
Example 3.
The Median being shown is from point D to point C.
A
B
C
D
What is the Centroid?
• CENTROID: the point of concurrency of the three medians of a triangle.
• The centroid is always in the inside of the triangle.
• The centroid of a triangle is located ⅔ of the distance from each vertex to the midpoint of the opposite side.
Example 1.
To find the centroid, bisect the angle.
Example 2.
Connect the angle bisector to the midpoint of the opposite side.
Example 3.
The point of intersection is the centroid of the triangle.
Concurrency of the Medians.
All of the Medians in a triangle are concurrent to each other.
Altitude of Triangles.
• Altitude of Triangles: a perpendicular segment from a vertex to the line containing the opposite side.
• Every single triangle has three altitudes.
• It can be inside, outside or on the triangle.
Example 1.
In this triangle the altitude is 4.5 centimeters.
Example 2.
In this triangle the altitude is 3 centimeters.
Example 3.
In this triangle the altitude is 11 centimeters.
What is the Orthocenter?
• ORTHOCENTER: the point of concurrency of the three altitudes of a triangle. • The concurrent altitudes
either intersect inside (acute), outside (obtuse) or on the vertex of the right angle.
Example 1.
Example 2.
Example 3.
What is a Midsegment?
• MIDSEGMENT: a segment that joins two midpoints of the sides of the triangles
• The midsegment is parallel to the opposite side.
• The Midsegment is half as long as the opposite side.
• THEOREM: a midsegment is parallel to a side of the triangle and it is half the length of that side.
Example 1.
The Midsegment shown here is from point D to point E.
Example 3.
It is 6 cm to the endpoints, because it lies on the Perpendicular Bisector
Example 3.
It is 6 cm to the endpoints, because it lies on the Perpendicular Bisector
Relationship between Longer and Shorter Sides and the Opposite Angles
• If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.
• If two angles of a triangle are not congruent, then the longer side side is opposite the larger angle.
• The smallest side is opposite to the shortest angle.
• The medium side is opposite to the medium angle
Example 1.
M<S > M<AThe larger angle is opposite to the longer isde.
AS
Example 2.
M<A > M<CAngle A is opposite from the largest side.
Example 3.
U
Angle A is opposite from the shortest side.
Exterior Angle Inequality.
• An exterior angle of a triangle is greater than either of the non-adjacent interior angles.
Example 1.
Example 2.
Example 3.
Triangle Inequality.
• The sum of any two side lengths of a triangle is great than the third side length
• The two sides are the two shortest lengths of the triangle.
• The third side is the longest side or hypotenuse.
Example 1.
It is a triangle because the sums of the legs are greater than the longest side.
Example 2.
It is a triangle because the sum of the sides are greater than the longest side.
Example 3.
It is not a full triangle because the sum of the shortest sides equal the longest side.
Indirect Proofs.
• In this type of proof you use logical argument.
STEPS TO FOLLOW:• Identify the conjecture that needs to be
proven.• Assume the opposite of the conclusion• Use direct reasoning to show that is leads
to a contradiction.• Conclude that the assumption is false, so
the original statement/conjecture must be true.
Example 1.
Statement:1. AD is perp to BC2. BDA is a straight
angle3. M<BDA = 180°4. <BDA is a right
angle5. M<BDA = 90°6. <BDA is not a
straight angle
Reason:1. Given2. Assume that
is it false3. Def. of St. Ang4. Def. of Perp.
Lines5. Def. of Rt. Ang6. Contradiction
A
B CD
Example 2.
Statement:1. Δ JKL is a right Δ2. ΔJKL is obstuse (<K)3. M<K + m<L = 90°4. M<K = 90° - m<L 5. M<K > 90°6. 90° - m<L >90°7. M<L <0°8. ΔJKL doesn’t have an
obstuse angle.
Reason: 1. Given2. Assume3. Acute <‘s of rt Δ
are comp.4. Subtract Prop5. Def. of obtuse6. Substitution7. Subtraction8. Conjecture is true
KJ
L
What is the Hinge Theorem?
• If two sides of one triangle are congruent to another pair of sides in another triangle, but the included angles are not congruent, then the longer third side is across the larger included angle.
• CONVERSE: The longer third side is across the larger included angle if the included angles of the triangles are not congruent even though they have two sides with the same length.
Example 1.
The measure of <A is less than the measure of <D, so EF is greater than BC.
A
B
C
D
E
F
Example 2.
The segment GI is greater than JL.
G
H
I
K
J
L
Example 3.
All of the angles in the right triangles are congruent, but PQ is greater than LM.
L
MN O
P
Q
Special Right Triangles
• 30-60-90: in a 30°-60°-90° triangle, the length is the hypotenuse is 2 times the length if the shorter leg, and the length if the longest side is the length if the shorter leg times √3.
• 45-45-90: in a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is the length if a leg times √2.
Example 1 (45-45-90).
x = 6√2
The hypotenuse has to be #√2.So the measure is 6√2.6
45°
x
Example 2 (45-45-90).
X = 5√2 ÷ 2
5X
5 = x√2 5 = x√25√2 x 2
Example 3 (45-45-90).
X = 8√2
45°
8
x
X = leg√2X = 8√2
Example 4 (30-60-90).
X = 9 and Y = 9√3
30°
60°
18x
y
18 = 2x9 = x
Y = x√3Y = 9√3
Example 5 (30-60-90).
X = 4√3 and Y = 8√3
60°
30°
yx
12
12 = x√312/√3 = x4√3 = x
Y = 2xY = 2(4√3) Y = 8√3
Example 6 (30-60-90).
Y = 16
60°
30°
8
8√3
y
Y = 2xY = 2(8)Y = 16
