Download Geometry - Semester 2

Geometry - Semester 2
Mrs. Day-Blattner
2/12/2016
Agenda 2/12/2016
1) Lesson 5: How can I fit a quadrilateral in a circle? Stations
ABCD 2) Problem Set 1-6 - corrections
3) Lesson 6. Experiments with Inscribed angles
4) Homework
5) Quiz - quick view
New vocabulary
In the first oval for lesson 5
“ Inscribed polygon”
Definition: A polygon is inscribed in a circle if all
the vertices (corners) of the polygon lie on the
circle.
Draw your own example and a non example
(draw a polygon that is NOT inscribed in a circle)
Station A.
Construct a rectangle such that all four
vertices of the rectangle lie on the circle
below.
Given a circle and a rectangle, what must
be true about the rectangle for it to be
possible to inscribe a congruent copy of it
in the circle?
Station A.
Given a circle and a rectangle, what must
be true about the rectangle for it to be
possible to inscribe a congruent copy of it
in the circle? The diagonals of the
rectangle have to be the length of the
diameter of the circle.
Station B.
Construct a kite in the circle below, and
explain the construction using geometry.
https://www.mathsisfun.com/definitions/kite.html
Textbook says on page 355:
A kite is a quadrilateral with
exactly two distinct pairs of
adjacent congruent sides.
Station C
The figure below shows a rectangle inscribed in a
circle.
a) List the properties of a rectangle.
b) List all the symmetries this diagram
possesses.
c) List the properties of a square.
d) List all the symmetries that a diagram of a
square inscribed in a circle possesses.
Station C
The figure below shows a rectangle inscribed in a
circle.
a) List the properties of a rectangle.
Opposite sides parallel and congruent
four right angles
diagonals congruent and bisect each other
a)Station C
b) List all the symmetries this diagram possesses.
Opposite sides are congruent
all four angles are congruent
diagonals are congruent
figure may be reflected onto itself across the perpendicular
bisector of the sides of the rectangle
-can rotate onto itself with either a 180 degree or 360 degree
rotation, clockwise or counterclockwise
Station C
c) List the properties of a square.
Opposite sides parallel
all sides congruent
four right angles
diagonals congruent, bisect each other, and are
perpendicular
Station C
d) List all the symmetries that a diagram of
square inscribed in a circle possesses.
Same as for rectangle PLUS
all four sides are congruent, figure may be
reflected onto itself across the diagonals, the
figure may be rotated onto itself with either 90
degree or 270 degree rotation, either clockwise
or counter clockwise
Station D
● A rectangle is inscribed into a circle (ABCD). The
rectangle is cut along one of its diagonals and
reflected across that diagonal to form a kite
(ABED). Draw the kite and its diagonal (AE). Find
all the angles in this new diagram, given that the
acute angle between the diagonals of the
rectangle in the original diagram was 40 degrees.
Problem Set
1. Using only a piece of 8.5 x 11 inch copy paper
and a pencil, find the location of the center of the
circle below.
Problem Set
1. Lay the paper across the circle so its corner lies
on the circle. The points where the two edges of
the paper cross the circle at the endpoints of the
diameter. Mark those points and draw the
diameter using the edge of the paper as a
straight edge. Repeat to get a second diameter.
Intersection of the two diameters is the center of
the circle.
Problem Set
2. Is it possible to inscribe a parallelogram that is
not a rectangle in a circle?
Problem Set
2. No, although it is possible to construct an
inscribed polygon with one pair of parallel sides (a
trapezoid) a parallelogram requires that both pairs
of opposite sides be parallel and both pairs of
opposite angles be congruent. Any parallelogram
that has opposite sides parallel and has vertices on
the circumference of the circle will have 90 degree
angles at the vertices and so the parallelogram will
always be a rectangle.
Problem Set
3. In the figure, BCDE is a rectangle inscribed in
Circle A. DE = 8, BE = 12, find AE.
AE is a radius.
Triangle CBE is a right triangle.
Using Pythagorean theorem
82 + 122 = CE2
64 + 144 = CE2
CE = (sqrt)(208) = sq.rt. (16x13) = 4 sq.rt.13
Problem Set
3. In the figure, BCDE is a rectangle inscribed in
Circle A. DE = 8, BE = 12, find AE.
AE is a radius.
CE = (sqrt)(208) = sq.rt. (16x13) = 4 sq.rt.13
AE = ½ CE = ½ (4) (sq.rt)13 = 2 sq.rt 13
Problem Set
4. Given the figure, BC = CD = 8 and AD = 13, find
the radius of the circle.
Mark the midpoint of BC as point E.
BE = EC = 4, So ED = 12.
Triangle EAD is a right triangle, so by Pythagorean
theorem, EA = sq.rt. (132 -122)= sq.rt (169 - 144)
Problem Set
Triangle EAD is a right triangle, so by Pythagorean
theorem, EA = sq.rt. (132 -122)= sq.rt (169 - 144)
EA = 5.
AC is a radius, so using Pythagorean theorem
again
52 + 42 = AC2
AC = sq.rt (25 + 16) = sq. rt (41)
Problem Set
5. In the figure DF and BG are parallel chords 14
cm apart. DF = 12 cm, AB = 10cm and EH is
perpendicular to BG.
Find BG.
Draw triangle DEA, measure of angle DEA = 90°
EH is a perpendicular bisector of DF and BG.
5. cont.
DE= 6cm
AD = 10cm (a radius)
By Pythagorean theorem EA = sq.rt.(102 - 62)
= sq.rt.(100 - 36)
= sq.rt (64) = 8 cm
5. cont.
In triangle ABH, angle AHB = 90°
AB = 10cm (a radius), AH = 6cm,
since EA + AH = 14 cm
and so BH = 8cm
Therefore, BG = 16cm.
6. Use the perpendicular bisectors of the sides of a
triangle to construct a circle that circumscribes the
triangle.
We did this before when we found the
circumcenter of 3 points.
(Return samples).
Lesson 6. Experiments with inscribed angles (p. 33)
Turn over.
6
Lesson 4. Experiments with Inscribed Angles.
Draw this diagram at the top of your page.
B
C
A
E
F
Use the diagram to help you identify the following
terms:
Arc
Minor and major arc
inscribed angle
central angle
intercepted arc of an angle
Use the diagram to help you identify the following
terms:
Arc
An arc is a portion of the circumference
of a circle
Use the diagram to help you identify the following
terms:
Minor and major arc
BE is a minor arc
EDB is a major arc
What’s the difference?
Use the diagram to help you identify the following
terms:
inscribed angle
Angle BDC is an inscribed angle
angle whose vertex is on the circle, and
each side of the angle intersects the
circle in another point.
Use the diagram to help you identify the following
terms:
Central angle
angle BAC is a central angle
An angle whose vertex is the center of
the circle.
Use the diagram to help you identify the following
terms:
intercepted arc of an angle
angle CDB and angle CAB both intercept
arc BC . What do you think it means for
an angle to intercept an arc?
Use the diagram to help you identify the following
terms:
intercepted arc of an angle
An angle intercepts an arc if the
endpoints of the arc lie on the angle, all
other points of the arc are in the interior
of the angle, and each side of the angle
contains an endpoint of the arc.
Homework.
Problem Set. (Lesson 4 6)
(page numbering on right after engage ny
S.25 - S.27)
1-7.