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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

2


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

3


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

4


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

5


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

43

1 2

Y FX

A

CB E

D

DY EF

AX BC

AB

DE=

AX

DY

YXF

A

CB E

D

6


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

7


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

8


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

9


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

10


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

11


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

12


5. Theorem: In a 45-45-90 triangle, the length of the hypotenuse is 2 times

the length of a leg.

a2a

13


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

14


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

15


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

16


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

17


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

18


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

19


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

20


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

21


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

22


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

23


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

24


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

25


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

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