|
8-1
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|
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.
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|
|
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8-1 (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.
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|
|
|
8-1 (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.
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|
|
|
8-2
|
|
In a right triangle, the
|
|
square of the hypotenuse is
|
|
equal to the sum of the
|
|
squares of the legs.
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|
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8-3
|
|
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.
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|
|
|
8-6
|
|
In a 45-45-90 triangle,
|
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the hypotenuse is √2 times as
|
|
long as a leg.
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|
|
|
8-7
|
|
In a 30-60-90 triangle,
|
|
the hypotenuse is twice as long
|
|
as the shorter leg, and the
|
|
longer leg is √3 times as long as
|
|
the shorter leg.
|
|
|
|
9-1
|
|
If a line is tangent to a
|
|
circle, then the line is
|
|
perpendicular to the
|
|
radius drawn to the
|
|
point of tangency.
|
|
|
|
9-1 (C1)
|
|
Tangents to a circle from
|
|
a point are congruent.
|
|
|
9-2
|
|
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.)
|
|
|
|
9-3
|
|
In the same circle or in
|
|
congruent circles, two
|
|
minor arcs are congruent if
|
|
and only if their central angles
|
|
are congruent.
|
|
|
|
9-4
|
|
In the same circle or in
|
|
congruent circles:
|
|
(1) Congruent arcs have
|
|
congruent chords.
|
|
(2) Congruent chords have
|
|
congruent arcs.
|
|
|
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9-5
|
|
A diameter that is
|
|
perpendicular to a chord
|
|
bisects the chord and its arc.
|
|
|
|
9-6
|
|
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).
|
|
|
|
9-7
|
|
The measure of an inscribed
|
|
angle is equal to half the measure
|
|
of its intercepted arc.
|
|
|
|
9-7 (C1)
|
|
If two inscribed angles intercept
|
|
the same arc, then the angles
|
|
are congruent.
|
|
|
9-7 (C2)
|
|
An angle inscribed in a
|
|
semicircle is a right angle.
|
|
|
|
9-7 (C3)
|
|
If a quadrilateral is
|
|
inscribed in a circle, then its
|
|
opposite angles are
|
|
supplementary.
|
|
|
|
9-8
|
|
The measure of an angle
|
|
formed by a chord and a tangent
|
|
is equal to half the measure
|
|
of the intercepted arc.
|
|
|
|
9-9
|
|
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.
|
|
|
|
9-10
|
|
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.
|
|
|
|
9-11
|
|
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.
|
|
|
|
9-12
|
|
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.
|
|
|
|
9-13
|
|
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.
|
|
