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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.