Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Scale invariance wikipedia , lookup
Engineering drawing wikipedia , lookup
Architectural drawing wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
List of works designed with the golden ratio wikipedia , lookup
Rational trigonometry wikipedia , lookup
Technical drawing wikipedia , lookup
Renormalization group wikipedia , lookup
Euclidean geometry wikipedia , lookup
Golden ratio wikipedia , lookup
Geometry UNIT 7 Proportions and Similarity Day 1 Ratio/Proportion and Means/Extremes Day 2 Similar Polygons Day 3 Similar Triangles Day 4 Parallel Lines and Proportional Parts Day 5 QUIZ REVIEW Day 6 QUIZ Day 7 Parts of Similar Triangles Day 8 Similarity Transformations Day 9 Scale Drawings and Models Review TEST Vocabulary Ratio Means Similar Polygons Dilation Enlargement Scale Drawing Extended Ratio Extremes Scale Factor Similarity Transformation Reduction Page 1 Proportion Cross Products Midsegment Center of Dilation Scale Model Ratio & Proportions WRITE & USE RATIOS A ratio is a comparison of two quantities using division. The ratio of quantities a and b can be expressed as a to b, a:b, or Example: a , where b 0. Ratios are usually expressed in simplest form. b SPORTS – A baseball player’s batting average is the ratio of the number of base hits to the number of at-bats, not including walks. According to ESPN MLB, Detroit Tigers Miguel Cabrera had the highest batting average in Major League Baseball in 2013. If he had 555 official at-bats and 193 hits, find his batting average. Round the answer to the nearest thousandth. Example: In Peru High School, if there are 73 teachers and 758 students. What is the approximate student-teacher ratio at our school? Extended ratios can be used to compare three or more quantities. The expression a:b:c means that the ratio of the first two quantities is a:b, the ratio of the last two quantities is b:c and the ratio of the first and the last quantities is a:c. Example: The ratio of the measures of the angles in ABC is 3:4:5. Find the measures of the angles. 3x° 5x° Page 2 4xo USE PROPERTIES OF PROPORTIONS An equation stating two ratios are equal is called a proportion. In the proportion a c , the b d numbers a and d are called the extremes of the proportion, while the numbers b and c are called the means of the proportion. a c b d The product of the extremes ad and the means bc are called cross products. THE PRODUCT OF THE MEANS = THE PRODUCT OF THE EXTREMES Example: CAR OWNERSHIP – Fernando conducted a survey of 50 students and found that 28 owned their own cars. If 755 students drive to his school, predict the number of students with their own cars. PERU Example: Solve for x: Example: Solve for x: 3x – 1 = 4 2x + 4 5 x 2 – 8x + 8 = 20 Page 3 -x + 2 5 PRACTICE: 1. 2. 3. 4. 5. Find the value of x. Page 4 6. Find the value of x. 7. 8. Find x and round to the nearest tenth if necessary. 9. Page 5 Similar Polygons People often customize their computer desktops using photos, centering the images at their original size or stretching them to fit the screen. This second method distorts the image, because the original and new images are not geometrically similar. Similar polygons have the same shape, but not necessarily the same size. (Also known as the similarity ratio.) If two polygons are similar, then their perimeters are proportional to the scale factor between them. The ratio of the lengths of the corresponding sides of two similar polygons is called the scale factor. The scale factor depends on the order of comparison. If a specific order is not given, scale factor is the ratio of the image to the pre-image. In the diagram, ABC ~ XYZ. A C Y The scale factor of ABC to XYZ is 6 or 2. 3 The scale factor of XYZ to ABC is 3 1 or . 6 2 a. b. Page 6 6 3 B X Z Example: Find the value of each variable if JLM ~ QST. J 4 S M 3 3y – 2 5 6x – 3 T 2 Q L Page 7 PRACTICE: 1. 2. 3. 4. Find the perimeter of: Page 8 5. **Determine if all angles are 90o by using slope formula and showing perpendicular lines. **Find the length of each side (distance formula) and determine if corresponding sides are proportional. Page 9 Similar Triangles Angle-Angle (AA) Similarity – If two angles of one triangle are similar to two angles of another triangle, then the triangles are similar. Example: Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. a. A C B b. J 47° P 44° L Example: Q K Find each measure. a) QP and MP b) WR and RT M S 5 Q x P 6 8 N W 3 3 5 x+6 2x + 6 R 10 V O Page 10 T Page 11 Page 12 Parallel Lines & Proportional Parts Triangle Proportionality Theorem – If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths. Example: In PQR, ST || RQ . If PS = 12.5, SR = 5 and PT = 15, find TQ. P T Q S R Converse of Triangle Proportionality Theorem – If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle. E Example: DG is half the length of GF, EH = 6, and HF = 10. Is DE || GH ? D H G F A midsegment of a triangle is a segment with endpoints that are the midpoints of two sides of the triangle. Every triangle has three midsegments. Triangle Midsegment Theorem – A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side. Page 13 Example: Find each measure. a) DE = _____ b) DB = _____ c) m FED = _____ A 15 F D 9.2 C 82° E B Proportional Parts of Parallel Lines – If three or more parallel lines intersect two transversals, then they cut off the transversals proportionately. Example: REAL ESTATE – Frontage is the measurement of a property’s boundary that runs along the side of a particular feature such as a street, lake, ocean, or river. Find the ocean frontage for Lot A to the nearest tenth of a yard. Ocean Lot A 60 yd. Lot B Congruent Parts of Parallel Lines – If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Example: Page 14 PRACTICE: 1. 2. 3. Page 15 4. 5. 6. Page 16 QUIZ REVIEW SHOW WORK for #1 – 5 below (if necessary): Page 17 SHOW WORK for #1-5 below (if necessary): Page 18 Parts of Similar Triangles Special Segments in Similar Triangles Label the following special segments as median, altitude, or angle bisector. Write a mathematical sentence below the label to prove you are correct! M A T S H S A W E O M E If two triangles are similar the lengths of the corresponding altitudes are proportional to the lengths of the corresponding sides. The same applies to angles bisectors and medians. Special note: In an equilateral triangle, the altitude, the angle bisector and the median are all the same line. In a scalene triangle, the altitude, the angle bisector and the median are all different lines. Try drawing all three lines on the triangles below. Example: In the figure, KLM ~ XZY. Find the value of x. K P L Y 20 16 M Page 19 x Z Q 15 X Triangle Angle Bisector – An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. Example: Find x. x 13 4 Example: 6 Find x. 14 11 x 20 Page 20 Page 21 PRACTICE: 1. 2. 3. Find x. Page 22 Similarity Transformations A transformation is an operation that maps an original figure, the preimage, onto a new figure called the image. A dilation is a transformation that enlarges or reduces the original figure proportionality. Since a dilation produces a similar figure, a dilation is a type of similarity transformation. Dilations are performed with respect to a fixed point called the center of dilation. The scale factor of a dilation describes the extent of the dilation. The scale factor is the ratio of a length on the image to a corresponding length on the preimage. The letter k usually represents the scale factor of a dilation. The value of k determines whether the dilation is an enlargement or a reduction. 3. Page 23 Page 24 PRACTICE: 1. 2. Page 25 Scale Drawings and Models A scale model or a scale drawing is an object or drawing with lengths proportional to the object it represents. The scale of a model or drawing is the ratio of a length on the model or drawing to the actual length of the object being modeled or drawn. Examples: Page 26 4. Page 27 Page 28 Page 29