81

If the altitude is drawn

to the hypotenuse of a

right triangle, then the two

triangles formed are

similar to the original

triangle and to each other.


81 (C1)

When the altitude is drawn

to the hypotenuse of a

right triangle, the

length of the altitude is the

geometric mean between

the segments of the hypotenuse.


81 (C2)

When the altitude is drawn

to the hypotenuse of a

right triangle, each leg is the

geometric mean between the

hypotenuse and the segment of

the hypotenuse that is

adjacent to that leg.


82

In a right triangle, the

square of the hypotenuse is

equal to the sum of the

squares of the legs.


83

If the square of one side of

a triangle is equal to the sum of

the squares of the other two

sides, then the triangle is a

right triangle.


86

In a 454590 triangle,

the hypotenuse is √2 times as

long as a leg.


87

In a 306090 triangle,

the hypotenuse is twice as long

as the shorter leg, and the

longer leg is √3 times as long as

the shorter leg.


91

If a line is tangent to a

circle, then the line is

perpendicular to the

radius drawn to the

point of tangency.


91 (C1)

Tangents to a circle from

a point are congruent.


92

If a line in the plane of a

circle is perpendicular to a

radius at its outer

endpoint, then the line is

tangent to the circle.


P16

The measure of the arc formed

by two adjacent arcs is the

sum of the measures of

these two arcs.

(Arc Addition Postulate.)


93

In the same circle or in

congruent circles, two

minor arcs are congruent if

and only if their central angles

are congruent.


94

In the same circle or in

congruent circles:

(1) Congruent arcs have

congruent chords.

(2) Congruent chords have

congruent arcs.


95

A diameter that is

perpendicular to a chord

bisects the chord and its arc.


96

In the same circle or in

congruent circles:

(1) Chords equally distant from the

centre (or centres) are congruent.

(2) Congruent chords are equally

distant from the centre (or centres).


97

The measure of an inscribed

angle is equal to half the measure

of its intercepted arc.


97 (C1)

If two inscribed angles intercept

the same arc, then the angles

are congruent.


97 (C2)

An angle inscribed in a

semicircle is a right angle.


97 (C3)

If a quadrilateral is

inscribed in a circle, then its

opposite angles are

supplementary.


98

The measure of an angle

formed by a chord and a tangent

is equal to half the measure

of the intercepted arc.


99

The measure of an angle

formed by two chords that intersect

inside a circle is equal to half

the sum of the measures of

the intercepted arcs.


910

The measure of an angle formed

by two secants,

two tangents,

or a secant

and a tangent

drawn from a point outside a circle

is equal to half the difference

of the measures of

the intercepted arcs.


911

When two chords intersect inside

a circle, the product of the

segments of one chord equals

the product of the segments of

the other chord.


912

When two secant segments are

drawn to a circle from an external

point, the product of one secant

segment and its external segment

equals the product of the other

secant segment and its

external segment.


913

When a secant segment and

a tangent segment are drawn to

a circle from an external point,

the product of the

secant segment and its

external segment is equal to

the square of the

tangent segment.

