# Modeling with Geometry

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1 Modeling with Geometry

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3 6.3 Parallelograms Properties of Parallelograms Sides A parallelogram is a quadrilateral with both pairs of opposite sides parallel. If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent. If a quadrilateral is a parallelogram, the 2 pairs of opposite angles are congruent. Angles If a quadrilateral is a parallelogram, the consecutive angles are supplementary. If a quadrilateral is a parallelogram and one angle is a right angle, then all angles are right angles. Diagonals If a quadrilateral is a parallelogram, the diagonals bisect each other. If a quadrilateral is a parallelogram, the diagonals form two congruent triangles. Example 1: Given: ABCD is a parallelogram. Prove: AB = CD and BC = DA. Statement Reason 1. ABCD is a parallelogram Definition of a parallelogram 3. <1 = <4, <3 = < AC = AC ABC = CDA CPCTC 1

4 Example 2: Given: ABCD is a parallelogram. Prove: AC and BD bisect each other at E. Statement Reason 1. ABCD is a parallelogram 1. Given 2. AB DC <1 = <4, <2 = < AB = DC ASA 6. AE = CE, BE = DE Definition of bisector Example 3: For what values of x and y must each figure be a parallelogram? a) b) c) d) e) f) 2

5 Homework 6.3: Parallelograms Math 3 Name: 1. Use the diagram below to solve for x and y if the figure is a parallelogram. a) PT = 2x, QT = y + 12, TR = x + 2, TS = 7y b) PT = y, TR = 4y -15, QT = x + 6, TS = 4x Find the measure of each angle if the figure is a rhombus. a) Find the m 1. b) Find the m 2. c) Find the m 3. d) Find the m Solve for x if the figure is a rhombus. 4. Solve for x if the figure is a rectangle. 5. What is the length of LN if the figure is a rectangle? 6. Solve for the missing angle measures if the figure is a rhombus. 7. What is the length of SW? 8. Solve for x if the figure is a rhombus. 3

6 6.4 Quadrilaterals Rectangle Rhombus Square A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides. A square is a parallelogram with four congruent sides and four right angles. A rectangle has all the properties of a parallelogram PLUS: 4 right angles Diagonals are congruent A rhombus has all the properties of a parallelogram PLUS: 4 congruent sides Diagonals bisect angles Diagonals are perpendicular A square has all the properties of a parallelogram PLUS: All the properties of a rectangle All the properties of a rhombus Example 1: Solve for x and the measure of each angle if DGFE is a rectangle. Example 2: ABCD is a rectangle whose diagonals intersect at point E. a) If AE = 36 and CE = 2x 4, find x. b) If BE = 6y + 2 and CE = 4y + 6, find y. Example 3: Using the diagram to the right to answer the following if ABCD is a rhombus. a) Find the m 1. b) Find the m 2. c) Find the m 3. d) Find the m 4. Example 4: Solve for each variable if the following are rhombi. a) b) 4

7 Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides, called bases, and two nonparallel sides, called legs. Isosceles Trapezoids An isosceles trapezoid is a trapezoid with congruent legs. A trapezoid is isosceles if there is only: One set of parallel sides Base angles are congruent Legs are congruent Diagonals are congruent Opposite angles are supplementary Trapezoid Midsegment The median (also called the midsegment) of a trapezoid is a segment that connects the midpoint of one leg to the midpoint of the other leg. Theorem: If a quadrilateral is a trapezoid, then a) the midsegment is parallel to the bases and b) the length of the midsegment is half the sum of the lengths of the bases Example 5: CDEP is an isosceles trapezoid and m<c = 65. What are m<d, m<e, and m<f? Example 6: What are the values of x and y in the isosceles triangle below if DE DC? Example 7: QR is the midsegment of trapezoid LMNP. What is x and the length of LM? You Try! TU is the midsegment of trapezoid WXYZ. What is x and the length of TU? Kite A kite is a quadrilateral with two pairs of adjacent, congruent sides. Its diagonals are perpendicular. If a quadrilateral is a kite, then: Its diagonals bisect the opposite angles. One pair of opposite angles are congruent. One diagonal bisects the other. Example 4: Quadrilateral DEFG is a kite. What are m<1, m<2, and m<3? You Try! Quadrilateral KLMN is a kite. What are m<1, m<2, and m<3? 5

8 Homework 6.4: Quadrilaterals Directions: For questions #1-2, find the measure of each missing angle Directions: For questions #3-4, find x and the length of EF Directions: For questions #5-6, find the measures of the numbered angles in each kite Challenge Question: Solve for the unknown angle measures in the kite shown below. 6

9 Math Tangent Lines of Circles Unit 6 SWBAT solve for unknown variables using theorems about tangent lines of circles. Tangent to a Circle Ex: (AB) A line in the plane of the circle that intersects the circle in exactly one point. Ex: Segment AB is a tangent to Circle O. Point of Tangency The point where a circle and a tangent intersect. Ex: Point P is a point of tangency on Circle O. Tangent Theorem 1: Converse Theorem 1: If a line is tangent to a circle, then it is perpendicular to the radius draw to the point of tangency. If a line is perpendicular to the radius of a circle at its endpoint on a circle, then the line is tangent to the circle. Example: If RS is tangent, then PR RS. Example 1: Find the measure of x. a) b) Example 2: Find x. All segments that appear tangent are tangent to Circle O. a) b) Example 3: Is segment MN tangent to Circle O at P? Explain. 7

10 Tangent Theorem 2: If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. Example 4: Solve for x. To circumscribe is when you draw a figure around another, touching it at points as possible. Ex: The circle is circumscribed about the triangle. Circumscribed vs. Inscribed To inscribe is to draw a figure within another so that the inner figure lies entirely within the boundary of the outer. Ex: The triangle is inscribed in the circle. Tangent Theorem 3: (Circumscribed Polygons) When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to the circle. Example 5: Triangle ABC is circumscribed about O. Find the perimeter of triangle ABC. You Try! Find x. Assume that segments that appear to be tangent are tangent. a) b b) c) 8

11 Practice 6.7: Tangents of Circles Directions: Assume that lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? Directions: In each circle, what is the value of x to the nearest tenth? TY and ZW are diameters of S. TU and UX are tangents of S. What is msyz? Directions: Each polygon circumscribes a circle. What is the perimeter of each polygon?

12 6.8 Chords & Arcs of Circles Any segment with that are the center and a point on the circle is a. A that passes through the center is a of a circle. Example 1: Name the circle, a radius, a chord, and a diameter of the circle. Circle: Radius: Chord: Diameter: The given point is called the. This point names the circle. Any segment with that are on a circle is called a. Circle: Radius: Chord: Diameter: Since a is composed of two radii, then d = 2r and r = d/2 Theorem 1: Converse Theorem 1: Within a circle or in congruent circles, chords equidistant from the center or centers are congruent. Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers). Theorem 2: Converse Theorem 2: Within a circle or in congruent circles, congruent central angles have congruent arcs. Within a circle or in congruent circles, congruent arcs have congruent central angles. Theorem 3: Converse Theorem 3: Within a circle or in congruent circles, congruent central angles have congruent chords. Within a circle or in congruent circles, congruent chords have congruent central angles. Theorem 4: Converse Theorem 4: Within a circle or in congruent circles, congruent chords have congruent arcs. Within a circle or in congruent circles, congruent arcs have congruent chords. Example 2: The following chords are equidistant from the center of the circle. a) What is the length of RS? b) Solve for x. 10

13 Theorem 5: In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. Theorem 6: In a circle, if a diameter bisects a chord that is not a diameter, then it is perpendicular to the chord. Theorem 7: In a circle, the perpendicular bisector of a chord contains the center of the circle. Example 3: In O, CD OE, OD = 15, and CD = 24. Find x. Example 4: Find the value of x to the nearest tenth. You Try! Find the value of x to the nearest tenth. a) b) 11

14 Practice 6.8: Chords & Arcs of Circles 6. A student draws X with a diameter of 12 cm. Inside the circle she inscribes equilateral ABC so that AB, BC, and CA are all chords of the circle. The diameter of X bisects AB. The section of the diameter from the center of the circle to where it bisects AB is 3 cm. To the nearest whole number, what is the perimeter of the equilateral triangle inscribed in X? 12

15 6.9 Inscribed Angles Major Arc: Minor Arc: Semicircle: An arc of a circle measuring more than or equal to 180 An arc of a circle measuring less than 180 An arc of a circle measuring 180 Central Angle: Central Angle Theorem: A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle. In a circle, or congruent circles, congruent central angles have congruent arcs. Example 1: Identify the following in P at the right. For parts d-f, find the measure of each arc in P. a) A semicircle b) A minor arc c) A major arc d) ST e) STQ f) RT Inscribed Angle: Inscribed Angle Theorem: An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords. The measure of an inscribed angle is half the measure of its intercepted arc. Example 2: What are the values of a and b? You Try! What are the m A, m B, m C, and m D? 13

16 Corollary 1: Corollary 2: Corollary 3: Two inscribed angles that intercept the same arc are congruent. An angle inscribed in a semicircle is a right angle. The opposite angles of a quadrilateral inscribed in a circle are supplementary. Example 3: What is the measure of each numbered angle? a) b) You Try! Find the measure of each numbered angle in the diagram to the right. a) m 1 = b) m 2 = c) m 3 = d) m 4 = Tangent Chord Angle: An angle formed by an intersecting tangent and chord has its vertex "on" the circle. Tangent Chord Angle Theorem: The tangent chord angle is half the measure of the intercepted arc. Tangent Chord Angle = ½ (Intercepted Arc) Example 4: In the diagram, SR is tangent to the circle at Q. If mpmq = 212, what is the m PQR? You Try! In the diagram, KJ is tangent to O. What are the values of x and y? Practice: Find the value of each variable. For each circle, the dot represents the center

17 Homework 6.9: Inscribed Angles Math 3 Name: Directions: Find the value of each variable. For each circle, the dot represents the center Directions: Find the value of each variable. Lines that appear to be tangent are tangent Directions: Find each indicated measure for M. 10. mb 11. mc Directions: Find the value of each variable. For each circle, the dot represents the center

18 6.9 Angle Measures and Segment Lengths Theorem 1: Theorem 2: The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. Example 1: Find each measure. a) b) c) Example 2: Find the following angles. a) m MPN b) c) You Try! Find the following angles. a) b) x c) Arc AB 16

19 Theorem 3: For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle. Example 4: Find the value of the variable in O. a) b) c) You Try! What is the value of the variable to the nearest tenth? 17

20 Homework 6.9: Angles and Segments Directions: Solve for x Directions: Solve for each variable listed There is a circular cabinet in the dining room. Looking in from another room at point A, you estimate that you can see an arc of the cabinet of about 100. What is the measure of A formed by the tangents to the cabinet? Directions: Find the diameter of O. A line that appears to be tangent is tangent. If your answer is not a whole number, round to the nearest tenth Directions: CA and CB are tangents to variable. O. Write an expression for each arc or angle in terms of the given 11. using x 12. using y 13. mc using x 18

21 6.10 Equations of Circles Standard Form of Circles Center: Radius: Point on the circle: Example 1: Write the equation of a circle with the given information. a) Center (0,0), Radius=10 b) Center (2, 3), Diameter=12 h = k = r = h = k = r = Example 2: Determine the center and radius of a circle the given equation b) ( x 3) ( y 5) 81 a) x y c) ( x 4) ( y 6) 1 Example 3: Use the center and the radius to graph each circle a) ( x 2) y 64 b) x ( y 4) 36 Center: Radius: Center: Radius: Writing an Equation with a Pass-Thru Point Example 4: Write the equation of a circle with a given center (2, 5) that passes through the point (5,-1). Step 1: Substitute the center (h, k) into the equation Step 2: Substitute the pass through point (x, y) into the equation for x and y. Step 3: Simplify and solve for r 2. Step 4: Substitute r 2 back into the equation from Step 1. 19

22 Writing an Equation with Two Points on the Circle Midpoint Formula Find the midpoint (radius) between the two endpoints, and then follow steps 1-4. Example 5: Write the equation of a circle with endpoints of diameter at (-6, 5) and (4, -3). Writing the Equation of a Circle in Standard Form Step 1: Step 2: Step 3: Step 4: Group x s and group y s together. Move any constants to the right side of the equation. Use complete the square to make a perfect square trinomial for the x s and then again for the y s. *Remember, whatever you do to one side of the equation, you must do to the other! Simplify factors into standard form of a circle! Example 5: Write the equation of a circle in standard form. Then, state the center and the radius. a) x 2 + y 2 + 4x - 8y + 16 = 0 b) x 2 + y 2 + 6x - 4y = 0 c) x 2 + y 2-6x - 2y + 4 = 0 d) x 2 + y 2 + 8x - 10y - 4 = 0 20

23 Homework 6.10: Equations of Circles Note: If r 2 is not a perfect square then leave r in simplified radical form but use the decimal equivalent for graphing. Example: ) Graph the following circle: a. (x - 3) 2 + (y + 1) 2 = 4 b. (x 2) 2 + (y 5) 2 = 9 c. (y + 4) 2 + (x + 2) 2 = 16 2) For each circle, identify its center and radius. a. (x + 3) 2 + (y 1) 2 = 4 b. b. x 2 + (y 3) 2 = 18 c. (y + 8) 2 + (x + 2) 2 = 72 Center: Radius: Center: Radius: Center: Radius: 3) Write the equation of the following circles: 4) Give the equation of the circle that is tangent to the y-axis and center is (-3, 2). 5) Compare and contrast the following pairs of circles a. Circle #1: (x - 3) 2 + (y +1) 2 = 25 Circle #2: (x + 1) 2 + (y - 2) 2 = 25 b. Circle #1: (y + 4) 2 + (x + 7) 2 = 6 Circle #2: (x + 7) 2 + (y + 4) 2 = 36 21

24 6) Find the standard form, center, and radius of the following circles: a. x 2 + y 2 4x + 8y 5 = 0 b. 4x 2 + 4y y + 5 = 0 Center: Radius: Center: Radius: 7) Graph the following circles. a. x 2 2x + y 2 + 8y 8 = 0 b. x 2 + y 2 6x + 4y 3 = 0 8) Give the equation of the circle whose center is (5,-3) and goes through (2,5) 9) Give the equation whose endpoints of a diameter at (-4,1) and (4, -5) 10) Give the equation of the circle whose center is (4,-3) and goes through (1,5) 11) Give the equation whose endpoints of a diameter at (-3,2) and (1, -5) 22

25 Length of a Circular Arc Arcs have two properties. They have a measurable curvature based upon the corresponding central angle (measure of arc = measure of central angle). Arcs also have a length as a portion of the circumference. portion of circle whole circle central angle in degrees 360 central angle in radians 2 arc length circumference x length CB 360 2r -or - x (radians) 2 length CB 2r Remember: circumference of a circle = 2πr For a central angle θ in radians, and arc length s - the proportion can be simplified to a formula: 2 s 2r s2 2r s r Length of an Arc: for θ in radians s = rθ Examples: 1) For a central angle of π/6 in a circle of radius 10 cm, find the length of the intercepted arc. 2) For a central angle of 4π/7 in a circle of radius 8 in, find the length of the intercepted arc. 3) For a central angle of 40 in a circle of radius 6 cm, find the length of the intercepted arc. 4.) Find the degree measure to the nearest tenth of the central angle in a circle that has an arc length of 87 and a radius of 16 cm. 23

26 Area of a Sector Sector of a circle: a region bounded by a central angle and the intercepted arc Sectors have an area as a portion of the total area of the circle. portion of circle whole circle central angle in deg rees 360 central angle in radians 2 area of sector area of circle x 360 area of sector r 2 Remember: area of a circle = πr 2 -or - x (radians) 2 area of sec tor r 2 For a central angle θ in radians, and area of sector A, the proportion can be simplified to a formula: 2 A r 2 Area of a Circular Sector: A=½r 2 θ Examples: A2 r 2 A 1 2 r2 for θ in radians 5) Find the area of the sector of the circle that has a central angle measure of π/6 and a radius of 14 cm. 6) Find the area of the sector of the circle that has a central angle measure of 60 and a radius of 9 in. HONORS 7) A sector has arc length 12 cm and a central angle measuring 1.25 radians Find the radius of the circle and the area of the sector. 24

27 Practice: Arc Length & Area of Sectors 25

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29 Review: 1. Find x What is x? Find x. 9. Find x 10. Find HG 11. Find measure of arc x 12. Find CE 27

30 13. Which is the equation of a circle with r=11 and center (0,6)? 14. Find the arc length 15. Find x 16. Find the area of the sector 17. Find the center and radius 18. Find x 19. Find x and y 20. What is the length of RS? 21. Find WS 22. What is the measure of angle 1? 23. Find CA 24. Find x 28

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