•Objectives
•Develop and use the equation of a circle.
•Adjust the equation for a circle to move the center in a coordinate plane.
•The Equation of a Circle
•Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealing with points in the coordinate plane that are a given distance, r, from a single point, the center, that we can represent with a set of coordinates that we’ll call (h, k) so that we can use them anywhere in the plane.
•The Equation of a Circle, Cont.
•Since we are working with Distance, the Distance Formula will be used.
•The Equation of a Circle, Cont.
•An equation for a circle with center at (h, k), and a radius of r units is:
Thursday, April 24, 2008
Tuesday, April 22, 2008
Cirlce in coordinate plane.
Objectives:
- Develop and use the equation of a circle.
- Adjust the equation of the circle to move the center ub a coordinate plane.
The Equation of a Circle:
- Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealling with points in the coordinate plane that are a given distance, r, from a single point, the center, that we'll call (h,k) so that we can use them anywhere in the plane.
- since we are working with distance the distance formula will be used.
- Replace the letter with the new information we know.
- r= the square root of x-h squared + y - k squared.
- replaceing distance with the word radius.
- Square both sides and you'll get the radius squared=x-h squared + y-k squared
- an equation for a circle at (h,k) and a radius of r units is
- (x-h)squared+(y-k)squared=rsquared.
Key Skills:
Sketch a cirlce from its equation.
(x-2)2+(y-1)2=9
Monday, April 21, 2008
Thursday, April 17, 2008
Segments of Tangents, Secants and Chords
External Secant Segment: That part of a secant segment exterior to the circle
Secant Segment: That part of a secant thta extends from an external point that includes a chord of the cirlce.
Tangent Segment: That part of a tamgent that extends from an exterior point to the point of tangency.
Theorem 10-15: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
Theorem 9.5.2: If two secants intersect outside a circle. the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (Whole x Outside =Whole x Outside)
Secant Segment: That part of a secant thta extends from an external point that includes a chord of the cirlce.
Tangent Segment: That part of a tamgent that extends from an exterior point to the point of tangency.
Theorem 10-15: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
Theorem 9.5.2: If two secants intersect outside a circle. the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (Whole x Outside =Whole x Outside)
Wednesday, April 16, 2008
April 16th, 2008
Secant- Al line that intersects a circle in two points.
Secants create chords.
Interior Angle Theorem:
Definition: Angles that are formed by two intersecting chords.
The measure of the angle formed by two intersecting chords is equal to 1/2 the sum of the measure of the intercepted arcs.
Exterior Angles
An Angle formed by two secants and two tangents or a secant and a tangent drawn from a point outside the circle.
the measure of the angle formed is one 1/2 the distance of the intercepted arcs.
Secants create chords.
Interior Angle Theorem:
Definition: Angles that are formed by two intersecting chords.
The measure of the angle formed by two intersecting chords is equal to 1/2 the sum of the measure of the intercepted arcs.
Exterior Angles
An Angle formed by two secants and two tangents or a secant and a tangent drawn from a point outside the circle.
the measure of the angle formed is one 1/2 the distance of the intercepted arcs.
Tuesday, April 15, 2008
Notes for the 15th
tangent to a circle- a tangent us a line that touches a circle in one spot.
point of tangency- the single point where that line touches the circle.
Tangent Theorem 10-9: If a line is tangent to a circle then the line is perpendicular to a radius of the circle drawn to the point of tangency.
Theorem 10-10: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then hte line is a tangent to the circle
Theorem 10-11: If two segments from the same exterior point are tangent to a circle then they are congruent.
point of tangency- the single point where that line touches the circle.
Tangent Theorem 10-9: If a line is tangent to a circle then the line is perpendicular to a radius of the circle drawn to the point of tangency.
Theorem 10-10: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then hte line is a tangent to the circle
Theorem 10-11: If two segments from the same exterior point are tangent to a circle then they are congruent.
Monday, April 14, 2008
GEOMETRY NOTES 4/14
inscribed angle- an inscribed angle is an angle made up of two chords. Since it is made up of chords. The vertex of the angle will be ON the circle.
There are three types of inscribed angles center interior, center exterior, Center included.
Intercepted Arc- the mino acr defined by the two endpoints of chords forming an inscribed angle that are not part of the vertex of the inscribed angle.
Inscribed angle therorem 10-5: The measure if n angle inscribed in a circle is equal to half the measure of an intercepted arc (or, the measure of an intercepted arc is twice the measure of the inscribed angle)
Theorem 10-6: Arc Inctercept Theroem: If two inscribed angles intercept congruent arcs of the same arc the angles are congruent.
Theorem 10-7: Right-Angle: If an inscribed angle intercepts a semicircle. then the angle is a right angle.
There are three types of inscribed angles center interior, center exterior, Center included.
Intercepted Arc- the mino acr defined by the two endpoints of chords forming an inscribed angle that are not part of the vertex of the inscribed angle.
Inscribed angle therorem 10-5: The measure if n angle inscribed in a circle is equal to half the measure of an intercepted arc (or, the measure of an intercepted arc is twice the measure of the inscribed angle)
Theorem 10-6: Arc Inctercept Theroem: If two inscribed angles intercept congruent arcs of the same arc the angles are congruent.
Theorem 10-7: Right-Angle: If an inscribed angle intercepts a semicircle. then the angle is a right angle.
Sunday, April 13, 2008
PAGE 526 21-57
21 TUVWXZ
22 ZV
23 ZW
24 TX
25 ZV
26 4ft?
27 2.5 ?
28. 60cm.
29. 16in
30. 9
31. .06
32. 6
33. 8
34 6
37. 35
38. 8
39. 7
40. 15
41. 26
42. 18
41. 6
42. 12
43. 16
44. 5
45.6
46. 15
47. 19
48 16
49. 13
50. 18
51. 52
53. 14
54. 41
22 ZV
23 ZW
24 TX
25 ZV
26 4ft?
27 2.5 ?
28. 60cm.
29. 16in
30. 9
31. .06
32. 6
33. 8
34 6
37. 35
38. 8
39. 7
40. 15
41. 26
42. 18
41. 6
42. 12
43. 16
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48 16
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53. 14
54. 41
Tuesday, April 1, 2008
NOTES FOR APRIL 1ST, OH8
Position of The figure
-Position the figure to simplify the proof
Find the missing coordinate
-Opposite sides of a parallelogram are parallel & congruent, therefore the y coordinate od D must be C.
-The length of AB is a, and the length of DC is a, so rhe x coordinate of D must be (a+b)-a or b
-Thus the coordinate of D are (b,c)
If you are working with a parallelogram and you say each side top and bottom is a units long its easy for you to find this point, i'd say that it is C units long if the lower corner is a it mus be (a,0) because you didn't move around the distance. The parrallelogram is C units tall and you wanna find the top corner coordinate ( , ) so lower left hand corner is b so the coordinate corner of the dotted line is (b,c) so top corn is now ( ,c) how far did you move to get from there to there? b units so it is now (b,c) so the entire distance is (a+b,0) the one above is (a+b,c)
Coordinate Proof
Given a square ABCD with midpoints MNPQ
Proove MNPQ is a square.
How do we proove this?
4 congruent sides
4 congruent angles
prove its a rhombus
prove its a rectangle
then if it has this it is a square.
Prove that the diagnols are congruent.
Prove the diagnols are perpendicular.
-Position the figure to simplify the proof
Find the missing coordinate
-Opposite sides of a parallelogram are parallel & congruent, therefore the y coordinate od D must be C.
-The length of AB is a, and the length of DC is a, so rhe x coordinate of D must be (a+b)-a or b
-Thus the coordinate of D are (b,c)
If you are working with a parallelogram and you say each side top and bottom is a units long its easy for you to find this point, i'd say that it is C units long if the lower corner is a it mus be (a,0) because you didn't move around the distance. The parrallelogram is C units tall and you wanna find the top corner coordinate ( , ) so lower left hand corner is b so the coordinate corner of the dotted line is (b,c) so top corn is now ( ,c) how far did you move to get from there to there? b units so it is now (b,c) so the entire distance is (a+b,0) the one above is (a+b,c)
Coordinate Proof
Given a square ABCD with midpoints MNPQ
Proove MNPQ is a square.
How do we proove this?
4 congruent sides
4 congruent angles
prove its a rhombus
prove its a rectangle
then if it has this it is a square.
Prove that the diagnols are congruent.
Prove the diagnols are perpendicular.
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