STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks,

Transcription

1 STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY 3 rd Nine Weeks,

2 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents. Each nine weeks Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A section has also been developed to provide students with the opportunity to solve similar problems and check their answers. The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall Textbook Series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. To access the database of online resources scan this QR code. The Geometry Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. 2

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5 Polygons G.10 The student will solve real-world problems involving angles of polygons. Polygons A convex polygon is defined as a polygon with all its interior angles less than 180. This means that all the vertices of the polygon will point outwards, away from the interior of the shape. A non-convex (concave) polygon is defined as a polygon with one or more interior angles greater than 180. It looks like a vertex has been 'pushed in' towards the inside of the polygon. A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). 5

6 Interior and Exterior Angles n - # of sides Sum of measures of interior s Each Interior Angle Sum of the Exterior Angles Each Exterior Angle REGULAR POLYGONS IRREGULAR POLYGONS Will vary based on the algebraic or numerical expressions Supplementary to each of the corresponding interior angles Example 1: Given a regular nonagon (9 sided convex polygon), what are the following measures? a. The sum of the interior angles b. Each interior angle c. The sum of the exterior angle d. Each exterior angle Example 2: What are the values of x and y? a. 6

7 The interior angle, (5x + 5), and exterior angle, y, are supplementary. Therefore, Example 3: Each interior angle of a regular polygon is does the polygon have?. How many sides Polygons 1. What are the interior and exterior angle measures of a regular heptagon? 2. Given the 8-sided convex polygon, What is the value of n? 3. Each interior angle of a regular polygon is. How many sides does the polygon have? 7

8 Quadrilaterals G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. Properties of Quadrilaterals Quadrilateral Properties Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid Opposite Sides are Congruent Consecutive Angles are Supplementary Opposite Angles are Congruent Diagonals Bisect Each Other A parallelogram with 4 congruent sides Diagonals are perpendicular Each diagonal bisects opposite angles A parallelogram with 4 right angles Diagonals are congruent A parallelogram with 4 congruent sides and 4 right angles Exactly one pair of parallel sides Midsegment is parallel to bases Length of the midsegment is the average of the lengths of the bases Legs are congruent Base angles are congruent Diagonals are congruent Example 1: ABCD is a parallelogram, solve for y. Given: Diagonals of a parallelogram bisect each other. Therefore 8

9 Example 2: Based on the given information, can you prove that DEFG is a parallelogram? You can show that by Angle Side Angle. Because corresponding parts of congruent triangles are congruent you can show that. Once you show that both pairs of opposite sides are congruent, you can say that DEFG is a parallelogram. Example 3: Find the measure of the numbered angles in the rhombus = = = 58 3 = 4 4 = 58 2 = 32 The diagonals of a rhombus are perpendicular. Triangle Angle-Sum Theorem Alternate Interior Angles are Congruent Diagonals of a rhombus bisect opposite angles. Scan this QR code to go to a video tutorial on Quadrilaterals. Quadrilaterals 1. What value of x will make the figure at the right a rectangle? 2. Janet is making a garden in the shape of a rhombus. One pair of opposite angles each measure 70. What measure does each of the other opposite pair of angles measure? (Hint: Draw a picture.) 9

10 It is often easier to classify geometric figures when they are drawn in the coordinate plane. Using slopes, distances and midpoints can help you with this. Formula Example Find distance from A to B. A (-2, -1) B (6, 3) Distance Formula Find the midpoint of AB. A (-2, -1) B (6, 3) Midpoint Formula Find the slope of AB. A (-2, -1) B (6, 3) Slope Formula 10

11 Example 4: Is figure TRAP an isosceles trapezoid? In order to be an isosceles trapezoid, the legs must be the same length. Therefore must equal. Use the distance formula to determine if this is true. Find distance from T to P. T (-1, 3) P (-3, -2) Find distance from R to A. R (4, 3) A (5, -2) These distances are not the same therefore TRAP is NOT an isosceles trapezoid. Example 5: Is figure GRAM a square? To be a square we must show that all sides are the same length, and that all sides meet at right angles (are perpendicular to one another). Remember that for two sides to be perpendicular, their slopes must be negative reciprocals. Find the slope of AR. A (1, -3) R (6, 2) Find the slope of MA. M (-4, 2) A (1, -3) Find the slope of GR. G (1, 7) R (6, 2) Find the slope of GM. G (1, 7) R (-4, 2) All of the sides meet at right angles because 1 and -1 are negative reciprocals of each other. All of the sides will also have the same length ( ), therefore GRAM is a square. 11

12 Quadrilaterals Scan this QR code to go to a video tutorial on Coordinate Geometry. 3. What figure is formed by the points (-1, -3), (1, 2), (7, 3) and (5, 5) 4. What figure is formed by connecting the midpoints of figure RECT? Circles G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles. You name a circle by its center. This is Circle X (ʘ X). is a diameter is a radius is a chord is a central angle (an angle whose vertex is the center of a circle) is a semicircle (an arc that is half of a circle) is a minor arc (an arc that is less than a semicircle) is a major arc (an arc whose measure is greater than 180 (a semicircle)) You name a minor arc by its endpoints. You name a semicircle or major arc by its endpoints and another point on the arc. The measure of a minor arc is equal to the measure of its corresponding central angle. You can add adjacent arc measures to find the measure of combined arc. 12

13 Example 1: What is the measure of? because it is a semicircle therefore therefore You could have also found the measure of by (because ) The circumference of a circle is the measure of the distance around the outside of the circle. The formula for finding the circumference of a circle is Use the circumference along with arc measure to find the length of a given arc. Example 2: What is the length of, given? 13

14 Circles 1. Given that and are diameters of, find the measures of all minor arcs of. 2. Given that, find the length of. Express in terms of. 3. What is the circumference of? The area of a circle can be found using the formula. The sector of a circle is the region that is bounded by two radii. To find the area of the sector of a circle use the formula. Example 3: Find the area of sector BOC. Leave your answer in terms of. To find the area of a sector we will use the formula The measure of the arc is 90, and the radius is 6 in. 14

15 Sometimes you will be asked to find the area of a segment of a circle. A segment is made by joining the endpoints of an arc as shown in the picture below the shaded area is the segment of the circle. To find the area of the segment, use the radii from its endpoints to form a triangle. Example 4: Find the area of the shaded segment. Let s start by finding the area of the sector that includes the shaded segment. To find the area of the shaded region we need to subtract the area of the triangle from the area of the sector. Area of a triangle is found by the formula The base and height of this triangle are both 10.. The area of the shaded segment is the area of the sector with the area of the triangle subtracted. Scan this QR code to go to a video tutorial on Areas of Circles and Sectors. 15

16 In the picture below is tangent to. This means that is in the same plane as and intersects the circle in exactly one place. This place is point, and is called the point of tangency. If a line is tangent to a circle, then that line is perpendicular to the radius of the circle. Example 5: Is tangent to at? If is tangent to at then must be a right triangle. Use the Pythagorean Theorem to determine if is a right triangle. Side is Since is a right triangle, that means that is tangent to at. Scan this QR code to go to a video tutorial on Tangent Lines. 16

17 Circles 4. Find the area of the shaded region. Round to the nearest hundredth. 5. Find the radius of. Theorems about Chords and Arcs Within a circle, or in congruent circles, congruent central angles have congruent arcs. The converse is also true. If If, then., then. Within a circle or in congruent circles, congruent central angles have congruent chords. The converse is also true. If If, then., then. Within a circle or in congruent circles, congruent chords have congruent arcs. The converse is also true. If If, then, then. 17

18 Within a circle or in congruent circles, chords equidistant from the center or centers are congruent. The converse is also true. If If, then, then. In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. If is a diameter and Then and. In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord. If is a diameter and. Then. In a circle, the perpendicular bisector of a chord contains the center of a circle. If is the perpendicular bisector of Then contains the center of the circle. Example 6: Given, and. How can you show that Because the circles are congruent, you can say that their radii are congruent. Because the two congruent angles are across from these radii, you can say that the other angles across from the radii ( ) are also congruent. If you subtracted the two known angles from 180 you would have the angle measure of the central angle. These would have to be the same. 18

19 Theorems about Angles and Segments The measure of an inscribed angle is half the measure of its intercepted arc. The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measure of the intercepted arcs. The measure of an angle formed by two lines that intersect outside of a circle is half the difference of the measures of the intercepted arcs. 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 circle Case I Case II Case III Example 7: Find the value of each variable. Angle c is a vertical angle with the third angle in the triangle that includes s a and b. 19

20 Example 8: Find the value of x. Scan this QR code to go to a video tutorial on Angle Measures and Segment Lengths. Circles 6. What is the 7. What is the value of x? 20

21 Area and Perimeter The area of a regular polygon is found by the formula where a is the apothem (segment from center perpendicular to the side) and p is the perimeter. You may have to use what you know about interior angles of regular polygons, trigonometry and special right triangles to find these measures. Example 1: What is the area of a regular hexagon with a 4 in side? We know that a regular hexagon with 4 in sides has a perimeter of 24 in. Now we need the apothem. The apothem is. We can find the measure of by. This triangle is isosceles therefore and they are also 60. That means that right triangle is a triangle with. Using what we know about triangles, Example 2: What is the area of the regular pentagon shown below? We are given the apothem, but we need to use trig to solve for the side length. therefore. Now we can use trip to find. therefore The perimeter of this pentagon is and Scan this QR code to go to a video tutorial on Area and Perimeter. 21

22 Area and Perimeter 1. Find the area and perimeter of the figure below. 2. Find the shaded area of the regular hexagon. 3. What is the area of a regular octagon with side length 10 feet? Volume and Surface Area G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. Refer to the formula sheet for detailed pictures and symbols in formulas. The cone and pyramid have special symbols such as the slant height and perimeter of the base. 22

23 Scan this QR code to go to a video tutorial on Volume and Surface Area. Example 1: Find the volume and surface area of the square base pyramid pictured. I always start by writing the formulas that I need to use, and defining each of the variables. Volume: Surface Area: Example 2: The volume of a cylinder is. If the height is 14, solve for the radius. 23

24 Volume and Surface Area 1. What is the volume and surface area of the cylinder. 2. The surface area of a prism is 166 in². If the length is 4 and the width is 5, what is the height? Answers to the problems: Area and Perimeter Volume and Surface Area Polygons 1. Quadrilaterals trapezoid 4. rhombus Circles