Week of 10/31/2016

Geometry

Circles and Chords

A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.

Theorems:

1.2. 3.

In a circle, a radius perpendicular to a chord bisects the chord.In a circle, a radius that bisects a chord is perpendicular to the chord. In a circle, the perpendicular bisector of a chord passes through the center of the circle.

Proof of Theorem 1:

Statements		Reasons
1.		1.	Given
2.		2.	Two points determine exactly one line.
3.		3.	Perpendicular lines meet to form right angles.
4.		4.	A right triangle contains one right angle.
5.		5.	Radii in a circle are congruent.
6.		6.	Reflexive Property - A segment is congruent to itself.
7.		7.	HL - If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
8.		8.	CPCTC - Corresponding parts of congruent triangles are congruent.
9.	E is the midpoint of	9.	Midpoint of a line segment is the point on that line segment that divides the segment two congruent segments.
10.		10.	Bisector of a line segment is any line (or subset of a line) that intersects the segment at its midpoint.

 

Theorem:

In a circle, or congruent circles, congruent chords are equidistant from the center. (converse) In a circle, or congruent circles, chords equidistant from the center are congruent.

Theorem:

In a circle, or congruent circles, congruent chords have congruent arcs.(converse) In a circle, or congruent circles, congruent arcs have congruent chords.

;

Theorem:

In a circle, parallel chords intercept congruent arcs.

http://www.regentsprep.org/regents/math/geometry/gp14/circlechords.htm

Week of 10.31.2016-11.4.2016 College Readiness Math

Tutoring: Wednesday morning and Friday afternoon
Assessment: Quiz late in the week

No homework all week!

Monday
Happy Halloween1
Today, we finish up the questions with the scale drawing and look at other conversion problems.
Help:https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-geometry/cc-7th-scale-drawings/v/constructing-scale-drawings

Tuesday
It is the day to look at the conversion with height and weight of the students
Help:http://www.metric-conversions.org/length/feet-to-inches.htm


Wednesday
The day to look  at area, surface area and volume of figures and a review for the unit.
Help:http://www.math.com/tables/geometry/surfareas.htm


Thursday
 A prefinal exam assessment, you do not study for this assessment. We continue with volume today
Help:http://www.math.com/tables/geometry/volumes.htm


Friday
A little quiz to wrap up the measurement unit is today, I will run a quiz preview before the quiz.
Help: ( check any previous websites for the unit)

Have a great weekend!

Week of 10/24/2016

Geometry

Formulas for Angles in CirclesFormed by Radii, Chords, Tangents, Secants

Formulas for Working with Angles in Circles(Intercepted arcs are arcs "cut off" or "lying between" the sides of the specified angles.)

There are basically five circle formulas that you need to remember:

1.  Central Angle:  A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle. Central Angle = Intercepted Arc <AOB is a central angle.  Its intercepted arc is the minor arc from A to B. m<AOB = 80° Theorem involving central angles: In a circle, or congruent circles, congruent central angles have congruent arcs.

2.  Inscribed Angle:An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords. Inscribed Angle = Intercepted Arc <ABC is an inscribed angle.  Its intercepted arc is the minor arc from A to C. m<ABC = 50°           Special situations involving inscribed angles: An angle inscribed in a semi-circle is a right angle. In a circle, inscribed circles that intercept the same arc are congruent. A quadrilateral inscribed in a circle is called a cyclic quadrilateral. The opposite angles in a cyclic quadrilateral are supplementary.

3.  Tangent Chord Angle:An angle formed by an intersecting tangent and chord has its vertex "on" the circle. Tangent Chord Angle =  Intercepted Arc <ABC is an angle formed by a tangent and chord. Its intercepted arc is the minor arc from A to B. m<ABC = 60°

4.  Angle Formed Inside of a Circle by Two Intersecting Chords:When two chords intersect "inside" a circle, four angles are formed.  At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture.  Remember:  vertical angles are equal. Angle Formed Inside by Two Chords =  Sum of Intercepted Arcs Once you have found ONE of these angles, you automatically know the sizes of the other three by using your knowledge of vertical angles (being congruent) and adjacent angles forming a straight line (measures adding to 180). <BED is formed by two intersecting chords.  Its intercepted arcs are .   [Note:  the intercepted arcs belong to the set of vertical angles.]   also, m<CEA = 120° (vetical angle) m<BEC and m<DEA = 60° by straight line.

5.  Angle Formed Outside of a Circle by the Intersection of:"Two Tangents" or "Two Secants" or "a Tangent and a Secant". The formulas for all THREE of these situations are the same: Angle Formed Outside = Difference of Intercepted Arcs  (When subtracting, start with the larger arc.)

Two Tangents:<ABC is formed by two tangents intersecting outside of circle O. The intercepted arcs are minor arc  and major arc .  These two arcs together comprise the entire circle. Special situation for this set up:  It can be proven that <ABC and central <AOC are supplementary.  Thus the angle formed by the two tangents and its first intercepted arc also add to 180º.

Two Secants:<ACE is formed by two secants intersecting outside of circle O. The intercepted arcs are minor arcs  and .

a Tangent and a Secant:<ABD is formed by a tangent and a secant intersecting outside of circle O. The intercepted arcs are minor arcs  and .

Source: http://www.regentsprep.org/regents/math/geometry/gp15/circleangles.htm

Week of 10.24.2016 College Readiness Math

Tutoring Wednesday morning and Friday afternoon


Assessment : TBA

Monday
It is time to review the conversions via the cards and to create a real-world word problem

HW; None

Tuesday
We look at scale drawings
HW; None

Wednesday
You create a scale drawing of a marked off picture
HW; None


Thursday
It is measurement day, be ready to use a ruler
HW: Assessment in class on Friday


Friday
It is a mini test today on the conversion units and the scale drawing.
HW: None

Have a great weekend!