SIMILAR TRIANGLES - Amazon Web Services...SIMILAR TRIANGLES 1. Definition: A ratio represents the...
Transcript of SIMILAR TRIANGLES - Amazon Web Services...SIMILAR TRIANGLES 1. Definition: A ratio represents the...
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Acknowledgement: Scott, Foresman. Geometry.
SIMILAR TRIANGLES
1. Definition: A ratio represents the comparison of two quantities.
In figure, ratio of blue squares to white squares is 3 : 5
2. Definition: A proportion is an equation which states that two ratios are
equal.
The numbers 2 and 3 are proportional to 4 and 6.
3. Definition: Two convex polygons are similar if and only if there is a
correspondence between their vertices such that corresponding angles are
equal and corresponding sides are proportional.
=4
6
2
3
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4. Theorem: The ratio of the perimeter of two similar convex polygons is
equal to the ratio of lengths of any two corresponding sides.
5. Axiom: If two angles of one triangle are equal to two angles of another
triangle, the triangles are similar. AAA
6. Theorem: Similarity of triangles is reflexive, symmetric and transitive.
7. Theorem: If a line parallel to one side of a triangle intersects the other two
sides, then it divides them proportionally.
FB C
A
E
D
DE BC AD
DB =
AE
EC
E
B C
A
D
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8. Theorem: If a line not containing a vertex of a triangle cuts off on two of its
sides segments whose lengths are proportional to the lengths of these sides,
then this line is parallel to the third side of the triangle.
9. Theorem: If three parallel lines intersect two transversals, then the parallel
lines divide transversals proportionally.
DE BC AD
DB =
AE
EC
E
B C
A
D
AB
BC =
DE
EFAD BE CF
EB
C
A D
F
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10. Theorem: If an angle of one triangle is equal to an angle of a second
triangle, and if the lengths of the sides including these angles are
proportional, then the triangles are similar. SAS
11. Theorem: If the lengths of the sides of one triangle are proportional to the
lengths of the sides of a second triangle, then the triangles are similar. SSS
ABC DEF,AB
AC =
DE
DFA = D
FB C
A
E
AB
DE =
BC
EF =
AC
DF ABC DEF
FB C
A
E
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12. Theorem: In similar triangles, the lengths of bisectors of corresponding
angles are proportional to the lengths of corresponding sides.
13. Theorem: In similar triangles, the lengths of altitudes from corresponding
vertices are proportional to the lengths of corresponding sides.
1 = 2
AB
DE=
AX
DY 3 = 4
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1 2
Y FX
A
CB E
D
DY EF
AX BC
AB
DE=
AX
DY
YXF
A
CB E
D
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14. Theorem: In similar triangles, the lengths of medians from corresponding
vertices are proportional to the lengths of corresponding sides.
15. Theorem: The bisector of an angle of a triangle divides the opposite side
into two segments whose lengths are proportional to the lengths of the two
sides adjacent to the segments.
OR
The bisector of an angle of a triangle divides the opposite side in the ratio
and EY = YFBX = XCAB
DE=
AX
DY
YXF
A
CB E
D
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of the sides containing the angle.
16. Theorem: If an altitude is drawn to the hypotenuse of a right triangle, then
the new triangles formed are similar to the given triangle and to each other.
17. Definition: The geometric mean of two positive numbers a and b is the
positive number x such thatb
x
x
a .
Example: Suppose an investment of X earns 25% in the first year and 80% in the
1 = 2 AB
AC =
BD
DC
21
DB C
A
BAC BDA ADC
D
A
B C
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second year, then the average annual rate of return is 80.125.1 = 1.5 (geometric
mean of the rates of two years). Reference: When Less is More, MAA
18. Theorem: The length of the altitude to the hypotenuse of a right triangle is
the geometric mean of the lengths of the segments into which the altitude
separates the hypotenuse.
h
a b
c2 = ab
D
A
CB
b
h
a
a
h =
h
bh
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19. Theorem: If the altitude to the hypotenuse is drawn in a right triangle, then
the length of either leg is the geometric mean of the lengths of the
hypotenuse and the segment on the hypotenuse which is adjacent to the leg.
20. Theorem: The product of the lengths of the legs of a right triangle is equal
to the product of the lengths of the hypotenuse and altitude to this
hypotenuse.
AB2 = BD ∙ BC
D
A
B C
a
cb
h
bc = ah
D
A
CB
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PYTHAGORAS THEOREM
1. Theorem: In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
OR
In a right angled triangle, the square on the hypotenuse is equal to the sum
of the squares on the other two sides.
2. Theorem: If the sum of the squares of the lengths of two sides of a triangle
is equal to the square of the length of the third side, then the triangle is a
right triangle.
OR
If in a triangle, square on one side is equal to the sum of the squares on the
other two sides, then the angle opposite to the first side is right angle.
a
b c
c2 = a2 + b2
CB
A
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3. Theorem: In a 30-60-90 triangle, the length of the hypotenuse is twice the
length of the shorter leg, and the length of the longer leg is 3 times the
length of the shorter leg.
4. Theorem: The length of an altitude of an equilateral triangle with sides of
length s is s2
3.
a
2a3a
60°
A
C B
s3
2s
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5. Theorem: In a 45-45-90 triangle, the length of the hypotenuse is 2 times
the length of a leg.
a2a
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CIRCLES
1. Definition: A secant is a line that intersects a circle in two points.
2. Theorem: A line that lies in the plane of a circle and contains an interior
point of a circle is a secant.
3. Definition: A tangent to a circle is a line in the plane of the circle that
intersects the circle in exactly one point. The point of intersection is called
the point of tangency.
BA
O
t
O
P
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4. Definition: A sphere is the set of all points in space that are a given
distance from a given point. The given point is called the center of the
sphere.
A radius of a sphere is a segment determined by the center and a point on
the sphere.
A diameter of a sphere is a segment that contains the center and has its
endpoints on the sphere.
The intersection of a sphere and a plane containing the center of the sphere
is a great circle of the sphere.
Great Circles
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Small circle: Intersection of a plane containing an interior point of the
sphere, but not containing the center of the sphere.
5. Theorem: In a plane, a line is tangent to a circle if and only if it is
perpendicular to a radius drawn to the point of tangency.
OR
A tangent is perpendicular to the radius through the point of contact.
OR
A line drawn at the end of a radius perpendicular to it is a tangent to the
circle.
t
O
P
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6. Theorem: Segments drawn tangent to a circle from an exterior point are
equal.
7. Definition: A common tangent is a line that is tangent to each of two
coplanar circles.
Common external tangents do not intersect the segment joining the
centers of circles.
Common internal tangents intersect the segment joining the centers of
AB = AC
C
B
A
Common External Tangent
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circles.
8. Definition: Tangent circles are two coplanar circles that are tangent to the
same line at the same point.
9. Definition: A circle is circumscribed about a polygon when the vertices of
the polygon lie on the circle. The polygon is inscribed in the circle.
Common Internal Tangent
tangent
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10. Definition: A circle is inscribed in a polygon when the sides of the
polygon are tangent to the circle. The polygon is circumscribed about the
circle.
11. Definition: Concurrent lines are two or more lines that intersect in a single
point. The point is called the point of concurrency.
12. Definition: The circumcenter of a triangle is the point of concurrency of
the perpendicular bisectors of the sides of the triangle. The circumcircle is
k
mn
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the circumscribed circle.
13. Theorem: The angle bisector of a triangle are concurrent in a point
equidistant from the sides of the triangle.
14. Theorem: A circle can be inscribed in any triangle.
15. Definition: The incenter of a triangle is the point of concurrency of the
angle bisectors of the sides of the triangle. The inscribed circle is called
CIRCUMCENTER
B C
A
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incircle.
16. Definition: The orthocenter of a triangle is the point of concurrency of the
lines containing the altitudes the triangle.
Inscribed CircleGF
E
I
A
BC
F
H
E
DB C
A
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17. Definition: The centroid of a triangle is the point of concurrency of the
medians of the triangle.
18. If two chords intersect in a circle, then the product of the lengths of the
segments on one chord is equal to the product of the lengths of the
segments on the other.
1
2
D
G
F E
BC
A
= ODCOOBAO
O
D
C
A
B
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19. If two secant segments have a common endpoint in the exterior of a circle
then the product of the lengths of one secant segment and its external
segment equals the product of the lengths of the other secant segment and
its external segment.
20. If a secant segment and a tangent segment have a common endpoint in the
exterior of a circle then the product of the lengths of the secant segment
and its external segment is equal to the square of the length of the tangent
segment.
AO OB CO OD=
D
B
O
A
C
= PT2PBPA
BP
A
T
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AREA
1. Theorem: If two triangles are similar, then the ratio of their area is equal to
the square of the ratio of the lengths of any two corresponding sides.
2. Theorem: If two polygons are similar, then the ratio of their area is equal to
the square of the ratio of the lengths of any two corresponding sides.
3. Definition: A sector of a circle is a region bounded by two radii and either
the major arc or the minor arc that is intercepted.
area of ABC
area of DEF=
AB2
DE2 ABC DEF
FB C
A
E
D
O
A B
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4. Theorem: In a circle of radius r, the ratio of the length s of an arc to the
circumference C of the circle is the same as the ratio of the arc measure m
to 360.
360m
Cs
5. Definition: A segment of a circle is a region bounded by a chord and either
the major arc or the minor arc that is intercepted.
6. Definition: The ratio of the circumference C of a circle to the diameter is
denoted by .
s
C
m
O
A B
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Any two circles are similar. Hence
2
2
1
1
diameter
nceCircumfere
d
C
d
C
7. Theorem: A circle can be circumscribed about any regular polygon.
8. Theorem: A circle can be inscribed about any regular polygon.
9. Circumference of a circle is the limit of the perimeters of its inscribed
regular polygons as the number of sides increases.
Area of a circle is the limit of the area of its inscribed regular polygons as
the number of sides increases.
c2c1
d2d1