Download What is a circle?

Review
May 16, 2008
Right Triangles
• The altitude to the hypotenuse of a right
triangle divides the triangle into two triangles
that are similar to the original triangle and to
each other.
• AC is similar to CB
• AD is similar to CD
• CD is similar to DB
Pythagorean Theorem
• Pythagorean Theorem: In a right triangle, the
square of the hypotenuse is equal to the sum of
the squares of the legs.
• a2+b2=c2, where c is the hypotenuse.
45-45-90 Triangles
• 45°-45°-90° Theorem In a
45°-45°-90° triangle, the
hypotenuse is √2 times as
long as a leg.
• To move from a leg to the
hypotenuse, multiply by √2
( c = a√2)
• To move from the hypotenuse
to the leg, divide by √2 (a
= c/(√2)
Try it out… 45-45-90
• Find the missing values.
30-60-90 Triangles
• 30°-60°-90° Theorem 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.
• Where a is the short leg, b is the long leg, and c
is the hypotenuse…
• c= 2a, c=(b/√3)*2
• b=a√3, b= (c/2)*√3
• a= c/2, a= b/(√3)
c
b
a
Practice with 30-60-90
• Find the missing values.
What is a parallelogram?
• A parallelogram is a quadrilateral with both
pairs of opposite sides parallel.
Properties of parallelograms
•
•
•
•
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Diagonals bisect each other.
Five Ways to Prove a
Quadrilateral is a Parallelogram
1. Show that both pairs of opposite sides are parallel.
2. Show that both pairs of opposite sides are congruent.
3. Show that one pair of opposite sides are both parallel and
congruent.
4. Show that both pairs of opposite angles are congruent.
5. Show that the diagonals bisect each other.
What is a rectangle?
• A rectangle is a parallelogram with four right
angles.
Finding values for a
rectangle
3y+10
4x-12
3x
What is a rhombus?
• A rhombus is a parallelogram with four
congruent sides.
What is a square?
• A square is a parallelogram with four
congruent sides and four right angles.
What is a trapezoid?
• A quadrilateral with exactly one pair of parallel sides is called a
trapezoid. The parallel sides are called the bases and the other
sides are called the legs.
• Same side interior angles are supplementary.
Isosceles Trapezoids
• A trapezoid with congruent legs is called an isosceles trapezoid.
• If you cut out an isosceles trapezoid from a piece of paper and
fold it over so that the legs coincide, you will find the both pair
of base angles are congruent.
Median of a Trapezoid
• The median of a trapezoid is the segment
that joins the midpoints of the legs.
• The median of a trapezoid is very similar
to the midsegment of a triangle in that both
join the midpoints of sides.
• The median of a trapezoid…
– is parallel to the bases
– has a length equal to the average of the base
lengths (i.e. 1/2 the sum of the lengths of the
bases)
Using the median of
trapezoid
• What would you do to find x?
What is a circle?
• A circle is the set of points in a plane at a
given distance from a given point in that
plane. The given point is called the center of
the circle and the given distance is the radius.
• A circle is named for its center point.
• Any segment with one endpoint at the center of
a circle and the other on the circle is also
called a radius (we use the same term to
indicate a segment and its length). All radii
of a circle are congruent.
Other features of a circle
• A chord is a segment whose
endpoints lie on a circle.
• A secant is a line that
contains a chord.
• A diameter is a chord that
contains the center of a circle.
It is equal to twice the radius.
Tangents
• A tangent to a circle is a line in
the plane of a circle that
intersects the circle in exactly one
point, called the point of
tangency.
• We often say that rays and
segments are tangents if they
meet the criteria of the definition.
Congruent and Concentric
Circles
• Congruent circles are circles that have
congruent radii.
• Concentric circles are circles that lie in the
same plane and have the same center.
Properties of Tangents
• If a line is tangent to
a circle, then the line is
perpendicular to the
radius drawn to the
point of tangency.
• Line A is
perpendicular to AO.
More about properties of
tangents
• Tangents to a circle
from a point are
congruent.
• PA and PB are
congruent.
Central Angles
A central angle of a circle is an angle that has
its vertex at the center of the circle.
What is an arc?
An arc is an unbroken part of the circle. It is
measured in degrees. Its measure is equal to
the measure of its central angle
What is a major arc?
Measure of a major arc
• In the next diagram we see that the measure of
a major arc is 360 minus the measure of its
associated minor arc.
Semicircles
• A semicircle is an arc formed by a diameter.
It always measures 180 degrees. It is named
using three letters.
Adjacent Arcs
• Adjacent arcs of a circle are arcs that have exactly one point in
common. Arc AC and arc CB are adjacent arcs since they
share only the point C.
Arc Addition Postulate
• Arc Addition Postulate: The measure of the
arc formed by two adjacent arcs is the sum of
the measures of the two arcs.
Congruent Arcs
• Congruent arcs are arcs, in the same circle or
in congruent circles, that have equal measures.
Chords
• In the diagram below, chord AB cuts off two
arcs, arc AB and arc ATB.
• We call AB the minor arc, the arc of chord
AB.
More about chords
In the same circle or congruent
circles:
(1) Congruent arcs have
congruent chords
(2) Congruent chords have
congruent arcs.
Even more about chords..
• A diameter that is perpendicular to a chord
bisects the chord and its arc.
Just a little bit more
• In the same circle or in congruent circles:
(1) Chords equally distant from the center (or centers) are
congruent.
(2) Congruent chords are equally distant from the center (or
centers).
Inscribed angles
• An inscribed angle is an angle
whose vertex is on a circle and
whose sides contain chords of
the circle.
• The measure of an inscribed
angle is equal to half the
measure of its intercepted arc.
• Angle B = ½ of angle O
Inscribed angles with the same
arc
• If two inscribed angles intercept the same arc,
then the angles are congruent.
• Angle 1 = angle 2.
Inscribed in a semicircle
• An angle inscribed in a semicircle is a right
angle.
• Angle c = 90.
Inscribed quadrilateral
• If a quadrilateral is inscribed in a circle, then
its opposite angles are supplementary.
• X + Z = 180
• W + Y = 180
More about chords
• 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.
• Angle 1 = ½ (AC+BD)
Secants and tangent
• 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.
• 1 = ½ (x-y)
The relationship between
segments of chord
• Theorem: 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.
• r*s=t*u
Secant Segments
• Theorem: 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.
• r*s=t*u
Expanding to secant segments
and tangent segments
•r*s=t*u
Secant and tangent
• 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. (r*s= t2)