Geometry Honors - Chapter 7: Proportions and Similarity
Section 7.1 – Ratio and Proportions
· I can write and solve ratios and proportions.
· Ratio – a comparison of two quantities using division. It can be expressed 3 ways: a/b; a: b or a to b.
· Proportion – an equation stating that two ratios are equal. Use cross products or cross multiplying to solve. (“extremes” = “means”)
1. The number of students that participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth.
2. (a) In ∆EFG, the ratio of the measures of the angles is 5:12:13, and the perimeter is 90 centimeters. Find
the measures of the angles.
(b) In triangle the ratio of the measure of the sides is 3:3:8, and the Perimeter is 392 centimeters. Find
the measures of the sides.
3. Solve each problem:
4. Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat.
Homework – Page 460-463 (11 – 39 ODD and 38, 40, 42, 44, 46, 47, 48)
Section 7.2 – Similar Polygons
· I can use proportions to identify and solve problems of similar polygons.
· I can determine scale factor between two similar figures and use the scale factor to solve problems.
· Similar Polygons - figures that have the same shape, but not necessarily the same size. The angles will be congruent, but side lengths may be different. Similar polygons are represented by a ~. Ex: ABC ~ DEF
· Scale Factor – the ratio of the lengths of the corresponding sides of two similar polygons. It depends of the order of comparison. – Ex: page 466
1. If ∆ABC ~ ∆RST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.
2. Tan is designing a new menu for the restaurant where he works. Determine whether the following sizes for the new menu are similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning.
3. The two polygons are similar. (a) Find x and (b) Find y.
4.If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.
Homework – Page 469 – 472 (9 – 37 ODD and 28, 36, 40-45, 49, 51)
Section 7.3 – Similar Triangles
· I can identify corresponding sides and corresponding angles of similar triangles.
· I can demonstrate that corresponding angles and congruent and corresponding sides are proportional in a pair of similar triangles.
· I can determine that two figures are similar by verifying that angle measure is preserved and corresponding sides are proportional.
· I can show and explain that when two angles measures (AA) are known, the third angle measure is also known (Third Angle Theorem).
· I can use triangle similarity theorems such as AA, SSS, and SAS to prove two triangles similar.
· AA Similarity - If 2 angles of one triangle are congruent to 2 angles in another triangle, then the triangles are similar.
· SSS Similarity – If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
· SAS Similarity – If the lengths of 2 sides of one triangle are proportional to the lengths of 2 corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
1. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
2. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
3. If ∆RST and ∆XYZ are two triangles such that which of the following would be sufficient to prove that the triangles are similar?
4. Given RS ll UT, RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10. Find RQ and QT.
5. Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1P.M. The length of the shadow was 2 feet. Then he measured the length of Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?
Homework – Page 479-482 (9 – 23 ODD, 20, 22, 24, 29, 30, 31, 32, 41)
Section 7.4 – Parallel Lines and Proportional Parts
· I can prove the segment joining the midpoints of two sides of a triangle (midsegment) is parallel to, and half the length of, the third side.
· I can prove a line parallel to one side of a triangle divides the other two proportionally.
· I can prove if a line divides two sides of a triangle proportionally; then it is parallel to the third side.
· I can find the point on a line segment, given two endpoints, that divides the segment into a given ratio.
· 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. – Page 484 (ex: if BE ∥ CD, then (AB/BC) = (AE/ED))
· 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. Pg. 485
(Ex: If J and K are midpoints of FH and HG, then JK ∥FG and JK = 1/2FG)
1. In ∆RST, RT ll to VU, SV = 3, VR = 8, and UT = 12, find SU.
2. If ∆DEF, DH = 18, HE = 36, and DG = ½GF. Is GH ∥FE?
3. In the figure, DE and EF are midsegments of triangle ABC. Find each measure:
(a) AB (b) FE (c) m∠AFE
4.In the figure, Larch, Maple and Nuthatch Streets are all parallel. The figure shows the distances in between city blocks. Find x.
5. Find x and y.
Homework – Page 489-493 (11 – 27 ODD, 22, 26, 33, 34, 35, 37, 40, 41, 48, 49)
Section 7.5 – Parts of Similar Triangles
· I can recognize and apply proportional relationships of corresponding segments of similar triangles.
· I can find missing parts of triangles using the Angle Bisector Proportionality Theorem.
· Special segments of similar triangles –
o If 2 triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides.
o If 2 triangles are similar, the lengths of corresponding angle bisectors are proportional to the lengths of corresponding sides.
o If 2 triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides.
1. In the figure, triangle LJK ~ triangle SQR. Find the value of x.
2. Estimating Distance: Sanjay’s arm is about 9 times longer than the distance between his eyes. He sights a statue across the park that is 10 feet wide. If the statue appears to move 4 widths when he switches eyes, estimate the distance from Sanjay’s thumb to the statue.
3. Find x:
Homework – Page 499-501 (7 – 17 ODD, 10, 21 – 24, 31, 33)
Section 7.6 – Similarity Transformations & Section 9.6 - Dilations
· I can define similarity as a composition of rigid motions followed by dilations in which angle measure is preserved and side length is proportional.
· I can define dilation.
· I can perform a dilation with a given center and scale factor on a figure in the coordinate plane.
· I can verify that when a side passes through the center of dilation, the side and its image lie on the same line.
· I can verify that corresponding sides of the pre-image and images are parallel and proportional.
o Dilation – a transformation that enlarges or reduces to original figure proportionally.
o Dilation – with center ‘C’ and positive scale factor k, k ≠ 1, maps a point ‘P’ in a figure to its image such that:
o If point P and C coincide, then the image and preimage are the same point OR
o If point P is not the center of dilation, then P’ lies on CP and CP’ = k(CP)
o Center of dilation – a fixed point at which dilations are performed around.
o Scale factor of a dilation - describes the extent of the dilation. The ratio of a length on the image to the corresponding length on the preimage.
o Enlargement – an image larger than the original figure. It has a scale factor greater than 1.
o Reduction - an image smaller than the original figure. It has scale factor between 0 and 1.
1. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.
2. A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image?
3. (a) To create the illusion of a “life-sized” image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and puppet is 24 inches (LP). To what distance PP’ should you place the puppet from the screen to create a 49.5-inch tall shadow (I’M’) from a 9-inch puppet?
(b) Determine whether the dilation from Figure Q to Q’ is an enlargement or a reduction. Then find the scale factor of the dilation and x.
4. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation.
(a) Original: (-6, -3), (6, -3), (-6, 6) (b) Original: (2, 1), (4, 1), (2, 0), (4, 0) Image (-2, -1) (2, -1), (-2, 2) Image: (4, 2), (8, 2), (4, 0), (8,0)
5.. Trapezoid EFGH has vertices E(-8,4), F(-4,8), G(8,4) and H(-4,-8). Graph the image of EFGH after a dilation centered at the origin with scale factor of ¼.
Homework – Page 508-510 (6 – 17, 22, 27) ODD and Page 664 – 666 (15-25, 41)ODD
Section 7.7 – Scale Drawings and Models
· I can determine scale factor between two similar figures and use the scale factor to solve problems.
· Scale model or scale drawing – an object or drawing with lengths proportional to the object it represents.
· Scale – the ratio of a length on the model or drawing to the actual length of the object being modeled or drawn.
1. The distance between Boston and Chicago on a map is 9 inches. If the scale of the map is 1 inch: 95 miles, what is the actual distance from Boston to Chicago?
2. A miniature replica of a fighter jet is 4 inches long. The actual length of the jet is 12.8 yards.
(a) What is the scale of the model?
(b) How many times as long as the model is the actual jet?
3. Gerrard is making a scale model of his classroom on an 11- by- 17-inch sheet of paper. If the classroom is 20 feet by 32 feet, choose an appropriate scale for the drawing and determine the drawing’s dimensions.
Homework – Page 514-515 (5 – 14, 16, 17, 21)