1.5 segs __angle_bisectors
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1. A segment bisector splits a segment into two equal pieces. An angle bisector divides an angle into two congruent adjacent angles. 2. The midpoint of a segment is the point that bisects it. The midpoint formula can be used to find the coordinates of the midpoint between two points. 3. Compass and straightedge are tools used for geometric constructions. An example shows finding the midpoint and determining an unknown endpoint of a segment using the midpoint formula.
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An ellipse is the locus of a point which moves in such a way that its distance form a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus and fixed line is called the directrix and the constant ratio is called the eccentricity of a ellipse denoted by (e).
In other word, we can say an ellipse is the locus of a point which moves in a plane so that the sum of it distances from fixed points is constant.
2.1 Standard Form of the equation of ellipse
Let the distance between two fixed points S and S' be 2ae and let C be the mid point of SS.
Taking CS as x- axis, C as origin.
Let P(h,k) be the moving point Let SP+ SP = 2a (fixed distance) then
(ii) Major & Minor axis : The straight line AA is called major axis and BB is called minor axis. The major and minor axis taken together are called the principal axes and its length will be given by
Length of major axis 2a Length of minor axis 2b
(iii) Centre : The point which bisect each chord of an ellipse is called centre (0,0) denoted by 'C'.
(iv) Directrix : ZM and Z M are two directrix and their equation are x= a/e and x = – a/e.
(v) Focus : S (ae, 0) and S (–ae,0) are two foci of an ellipse.
(vi) Latus Rectum : Such chord which passes through either focus and perpendicular to the major axis is called its latus rectum.
Length of Latus Rectum :
If L is (ae, 𝑙 ) then 2𝑙 is the length of
SP+S'P=
{(h ae)2 k 2} +
= 2a
Latus Rectum.
Length of Latus rectum is given by
2b2
.
h2(1– e2) + k2 = a2(1– e2)
Hence Locus of P(h, k) is given by. x2(1– e2) + y2 = a2(1– e2)
2
a
(vii) Relation between constant a, b, and e
a 2 b2
b2 = a2(1– e2) e2 =
a 2
x2
a 2 +
y
a 2 (1 e 2 ) = 1
e =
a 2
Result :
Major Axis
(a) Centre C is the point of intersection of the axes of an ellipse. Also C is the mid point of AA.
(b) Another form of standard equation of ellipse
x 2 y2
a 2 b2
1 when a < b.
Directrix Minor Axis Directrix x = -a/e x = a/e
Let us assume that a2(1– e2 )= b2
The standard equation will be given by
x2 y2
a2 b2
2.1.1 Various parameter related with standard ellipse :
In this case major axis is BB= 2b which is along y- axis and minor axis is AA= 2a along x- axis. Focus S(0,be) and S(0,–be) and directrix are y = b/e and y = –b/e.
2.2 General equation of the ellipse
The general equation of an ellipse whose focus is (h,k) and the directrix is the line ax + by + c = 0 and the eccentricity will be e. Then let P(x1,y1) be any point on the ellipse which moves such that SP = ePM
Let the equation of the ellipse x
y2
a > b
(x –h)2 + (y –k)2 =
e 2 (ax1 by1 c) 2
a 2 b2
1 1 a 2 b2
(i) Vertices of an ellipse : The point of which ellipse cut the axis x-axis at (a,0) & (– a, 0) and y- axis at (0, b) & (0, – b) is called the vertices of an ellipse.
Hence the locus of (x1,y1) will be given by (a2 + b
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1.5 segs __angle_bisectors
- 1. 1.5 Segment & Angle Bisectors p.34
- 2. Always Remember! • If they are congruent, then set their measures equal to each other!
- 3. Midpoint • The point that bisects a segment. • Bisects? splits into 2 equal pieces 12x+3 10x+5 A M B 12x+3=10x+5 2x=2 x=1
- 4. Segment Bisector • A segment, ray, line, or plane that intersects a segment at its midpoint. k A M B
- 5. Compass & Straightedge • Tools used for creating geometric constructions • We will do a project with these later.
- 6. Midpoint Formula • Used for finding the coordinates of the midpoint of a segment in a coordinate plane. • If the endpoints are (x1,y1) & (x2,y2), then x1 + x2 y1 + y2 , 2 2
- 7. Ex: Find the midpoint of SP if S(-3,-5) & P(5,11). − 3 + 5 − 5 +11 , 2 2 2 6 , 2 2 (1,3)
- 8. Ex: The midpoint of AB is M(2,4). One endpoint is A(-1,7). Find the coordinates of B. x1 + x2 y1 + y2 , = (midpoint ) 2 2 x1 + x2 y1 + y2 =2 =4 2 2 − 1 + x2 7 + y2 =2 =4 2 2 − 1 + x2 = 4 7 + y2 = 8 x2 = 5 y2 = 1 ( 5,1)
- 9. Angle Bisector • A ray that divides an angle into 2 congruent adjacent angles. A D BD is an angle bisector of <ABC. B C
- 10. Ex: If FH bisects <EFG & m<EFG=120o, what is m<EFH? E 120 H = 60 o 2 F G m < EFH = 60 o
- 11. Last example: Solve for x. * If they are congruent, set them equal to each other, then solve! o 0 x +4 x+40=3x-20 o 3x-20 40=2x-20 60=2x 30=x
- 12. Assignment