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

a

b

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

Parallelogram1. 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

3x

4x-12

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)