Unit 8: Similarity. Part 1 of 2: Intro to Similarity and Special Proportions

Transcription

1 Name: Geometry Period Unit 8: Similarity Part 1 of 2: Intro to Similarity and Special Proportions In this unit you must bring the following materials with you to class every day: Please note: Calculator Pencil This Booklet A device Headphones! You may have random material checks in class Some days you will have additional handouts to support your understanding of the learning goals in that lesson. Keep these in a folder and bring to class every day. All homework for part one of this unit is in this booklet. Answer keys will be posted as usual for each daily lesson on our website

2 Today s Goals: 8-1 Notes * What does it mean for two figures to be similar? * What proportional relationships exist between the corresponding sides of similar figures? * What ratios exist between corresponding sides and perimeters in similar figures? * What ratios exist between corresponding sides and areas in similar figures? Compare it! Put a "C" over the pairs you think are congruent Put an "S" over the pairs you think are similar Think out Loud: What do you think it means for geometric figures to be "Similar"? Big Idea: Similar Figures Definition: figures whose are and whose are in Example: Notation: Proportionality in Similar Figures If figures are similar, then we can set up proportions using their corresponding sides

3 Identifying Similar Figures and Corresponding Parts Recall: Corresponding sides and angles are a pair of angles or sides that are in the same relative spot in two different shapes. If Δ ABC Δ DEF let s identify the pairs of corresponding angles and sides. Corresponding Sides Corresponding Angles AB is in proportion with is in proportion with DF is in proportion with Writing Ratios: 1) If the following figures are similar, determine the ratio of the sides (in simplest form) and the length of the missing side. a) b)

4 2) Are the following two triangles similar? Use ratios to justify your answer 1. Check ALL 3 Ratios This response requires 2 parts 2. Conclude One More Relationship Consider the following figure A and Figure B: Figure Corresponding Sides Ratio Ratio of Perimeter Ratio of Area A B Generalize: Perimeter vs. Sides The ratio of the perimeters is equal to the ratio of Area vs. Sides The ratio of the areas is equal to the ratio of

5 Let s Try Some! 3) If the ratio of the sides of two triangles is 3:2, what is the ratio of their perimeters? What would be the ratio of their areas? 4) The lengths of two corresponding sides of two similar triangles are 9 inches and 6 inches. If the perimeter of the smaller triangle is 24 inches, find the length of the perimeter of the larger triangle. 5) If the sides of a triangle are 15, 25, and 30 and the perimeter of a similar triangle is 28, find the length of the shortest side.

6 8-1 Homework Complete each of the following problems. Show all work and remember to check your answers! 1. Fill in the missing sides to the triangles below given that ABC is similar to HGF AB is corresponding to BC is corresponding to CA is corresponding to 2) If the ratio of the sides of two triangles is 3:7, A) What is the ratio of their perimeters? B) What is the ratio of their areas? 3) The lengths of two corresponding sides of two similar triangles are 18 inches and 16 inches. If the perimeter of the smaller triangle is 40 inches, find the length of the perimeter of the larger triangle. When in doubt, draw it out! 4) If the sides of a triangle are 2, 5, and 6 and the perimeter of a similar triangle is 39, find the length of the longest side.

7 Watch the following video to prepare for our next lesson. You will be quizzed on the content of this video tomorrow when you get to class. Let s Re-Activate our Knowledge: In an isosceles triangle, angle bisectors, perpendicular bisectors,, and all coincide. For two figures to be similar, their corresponding sides must be and corresponding angles must be. To prove triangles are similar we would have to show all 3 corresponding sides are in proportion AND all 3 pairs of corresponding angles are congruent! OR *It turns out, that when only some of the conditions exist to prove triangles similar, all of the conditions exist! to Proving Triangles Similar Method 1 Theorem: Method 2 Theorem: Method 3 Theorem: Two pairs of corresponding are. All three pairs of of corresponding are by the same. Two pairs of corresponding are and one pair of corresponding are. Quick Practice - Proving Triangles are Similar 1) Show that the following two triangles are similar. 2) Are the following two triangles similar? Be sure to justify your work! Justify your response.

8 8-2 Notes 8-2: Similarity Ratios Today s Goals: (1) What ratios exist between sets of corresponding sides and segments in similar triangles? (2) What are the minimum requirements to prove 2 triangles are similar? Goal Question: What ratios exist between sets of corresponding sides and special segments in similar triangles? From ticket in, we know that by 1. What is the ratio of corresponding sides? 2. How would you classify by their sides? 3. Sketch in altitudes ED and altitude HM 4. Because the two triangles are, the altitude you just sketch in is also the: 5. Solve for the length of ED. 6. Solve for the length of HM. 6. Write the ratio that exists between the special segments between each triangle. 7. What conclusion can you make about the relationship that exists between the ratio of the corresponding sides and the ratio of the special segments?

9 In similar triangles, the ratio between corresponding sides and are. So, if the ratio of the sides of two triangles is 4:5, what is the ratio of their Altitudes? Medians? Perpendicular bisectors? Angle Bisectors? Areas? You try it! The lengths of two corresponding sides of two similar triangles are 9 inches and 6 inches. If an altitude of the smaller triangle is 4 inches, find the length of the corresponding altitude of the larger triangle. Plan Work Time to Practice the Concepts from Today s Lesson! Directions: Complete the following problems. Remember to draw sketches to visual the figures you are working with in each example. 1. Given, determine the corresponding sides and solve for x. 2. If the ratio of the sides of two triangles is 3:2, what is the ratio of their areas?

10 3. Corresponding angle bisectors of two similar triangles have lengths of 18 millimeters and 12 millimeters. If the length of the median of the larger triangle is 48 millimeters, what is the length of the median of the smaller triangle? 4. The lengths of a pair of corresponding sides of two similar triangles are 6 inches and 5 inches. If the length of the altitude of the bigger triangle is 24 inches, what is the length of the altitude of the smaller triangle? 5. Two similar triangles have areas of 144 square centimeters and 64 square centimeters. a) Write the ratio of the areas. (Keep in un-reduced form) b) Find the ratio of the lengths of two of their corresponding sides. c) Find the ratio of their altitudes.

11 6. The lengths of a pair of corresponding sides of two similar polygons are 6 inches and 5 inches. If the area of the smaller polygon is 20 square inches, find the area of the larger polygon. **Start by drawing a picture (of any polygon!), and then determining what you need to find! 7. The lengths of two corresponding sides of two similar triangles are 18 inches and 16 inches. If the perimeter of the smaller triangle is 40 inches, find the length of the perimeter of the larger triangle. 8. Given that a)what is the ratio of their corresponding medians? b) What is the ratio of their areas?

12 8-2 Homework 1) Use the lengths and angle markings to fill-in the blanks given that the two quadrilaterals are similar: MH is in proportion with A T What is the ratio of the AREAS of these quadrilaterals? 2) In the accompanying diagram, is similar to,,,, and. a) Based on the given information, which side of corresponds to QS? b) What is the ratio of the corresponding sides? What proportion could be used to find the length of MN? 4) A triangle has sides whose lengths are 5, 12, and 13. A similar triangle could have sides with lengths of (1) 3, 4, and 5 (2) 6, 8, and 10 (3) 7, 24, and 25 (4) 10, 24, and 26 5) Corresponding altitudes of two similar triangles have lengths of 8 and 6. If the length of the median of the larger triangle is 20, what is the length of the median of the smaller triangle?

13 6) As shown in the diagram below,,,,, and. What is the length of? What is the ratio of the corresponding altitudes? What is the ratio of the areas? 7) The lengths of two corresponding sides of two similar triangles are 3 inches and 4 inches. If the perimeter of the larger triangle is 12 inches, find the length of the perimeter of the smaller triangle. 8) Justify that triangle ADE is similar to triangle ABC. Re-draw diagram to see the two triangles clearly!

14 8-3 Notes Lesson 8-3: Similar Triangles formed by Parallel Lines and Midsegment Theorem for Triangles and Trapezoids Learning Goal #1: How can similarity help us to find segment lengths when a line is drawn parallel to one side of a triangle? Pair Exploration PART 1: With your neighbor. Read through and answer the following questions. 1) How many triangles are in the diagram below? Name them. What do we know about sides and and how do we know? 2) Do you see any congruent angles in both triangles? How do you know they are congruent? 3) Based on the work we ve done in the last couple of lessons, what can you conclude about and? Together! Based on the following diagram, we can conclude that when one or more parallel lines are drawn to any side of a triangle then the triangles must be. Let s Try it! Example 1: Use a proportion to solve for.. Steps THIS IS THE MOST IMPORTANT STEP 1. Work 2.

15 Learning Goal #2: What is a midsegment and how is this segment related to similarity? Pair Exploration- Part 2- Midsegments. READ FIRST! Given: D is a midpoint of, E is a midpoint of. The segment connecting these midpoints is called the MIDSEGMENT Re-draw the two triangles created by the midsegment and label what you know (congruent angles or size lengthsdon t forget reflexive). 2. Are the two pairs of labeled corresponding sides proportional? Prove mathematically! 3. Based on the information from questions #1-2, we can conclude that is to by the method. 4. Since the triangles are similar, then ALL three pairs of corresponding sides are, what would be the length of Together! Let s see if we can find a relationship that occurs when midsegments are drawn in any triangle? BIG TAKE AWAY! Midsegments in triangles are always the length and also parallel to the side opposite them.

16 WATCH OUT FOR THIS COMMON MISTAKE! Parallel segments in triangles are NOT ALWAYS MIDSEGMENTS; therefore the third side is not always half the length of the segment. For these, you MUST re-draw triangles and set up a proportion! EX: Length of BC is NOT 12 because DE is NOT a midsegment! *MIDSEGMENTS ARE ALWAYS, but parallel segments are not always. Let s Try it! 1) In the diagram below of ΔABC, DE is a midsegment of ΔABC, DE = 7, AB = 10, and BC = 13. Find the perimeter of ΔABC Practice! 2) In the diagram below of, D is the midpoint of, O is the midpoint of, and G is the midpoint of. If,, and, what is the perimeter of parallelogram CDOG? 1) 21 2) 25 Think! What type of segments are DO, OG and DG? 3) 32 4) 40

17 3) In, point D is on AB and point E is on BC such that DE AC. If DB =2, DA = 7 and DE = 3, what is the length of AC? 4) a) Re-draw the similar triangles including labels b) Solve for x. 5) In the diagram of below,,, and. Find the perimeter of the triangle formed by connecting the midpoints of the sides of.

18 8-3 Homework Complete the following problems. Re-draw triangles, when necessary, and show all work. 1) a) Based on the information given on the diagram, is a midsegment? Explain. b) Draw and label two similar triangles based on the diagram. b) Solve for x. What is the length of 2) In the triangle below is a midsegment of triangle ABC. If DE = 8, AB = 12, and BC is 15, determine the perimeter of triangle ABC. 3) Fill in the following blanks that will complete the given statements. (You might want to fill in the given information to the diagram). Look at the markings made on the triangle to help you!

19 Edpuzzle: VIDEO #1 Watch the assigned video and try your constructions on this page. Mastery of the content of this video is essential for our next lesson in class. Failure to watch the video will result in confusion and your inability to interact with your peers throughout the lesson. This page will be checked tomorrow in class and an entrance ticket into class will be assigned to prove your mastery of the concept. Special Case of Midsegment Theorem: TRAPEZOIDS Midsegment Theorem for a Trapezoid: The length of the of a trapezoid equals the of the two bases! Formula: 1) In the diagram below of trapezoid RSUT,, X is the midpoint of, and V is the midpoint of. If RS = 30 and XV = 44, what is the length of TU? 2. is the median (mid-segment) of trapezoid ABCD. EF = 25 and AD = 40. Solve for BC.

20 Factoring Sum and Product How to factor polynomials using sum and product: EdPuzzle: VIDEO #2 How to factor and solve polynomials using sum and product: Steps The standard form for a quadratic is ax 2 + bx + c = 0 1. We are looking for factors of C (numbers that multiply to the last number) that also have the sum of b (add to the middle number) 2. List your factors of c and identify which pair add to b 3. Set up your parenthesis to set up your factors appropriately Example Together ex 1) x x + 6 = 0 1. C = 6 Factors of 6: 1 & 6 2 & 3-1 & -6-2 & Which of those pairs add to 5? 2 & 3 3. (x+2)(x+3) = 0 x = -2 x = -3 Example Together You Try! ex 2) -12 -x + x 2 = 0 ex 3) x x + 10 = 0 Another Factoring Method Difference of two Squares 4) Solve: x 2-9 = 0

21 8-4 Notes Lesson Goal: How can similarity help us to find segment lengths when an altitude is drawn into a right triangle? Before we begin: FACT CHECK What type of segment is CD? How do you know? Mean Proportional Theorem: When an altitude is drawn in a right triangle and intercepts the hypotenuse, What are the key parts?

22 Shortcuts to Using the Mean Proportional Theorem Altitude Rule Leg Rule ***Used when you are solving for, or are given The ***Used when you are or for the altitude But, how am I supposed to remember those proportions???? What does the math teacher use to carry all of her similar triangles? Identifying Parts for the Shortcut Rules Which shortcut would we use here?

23 Example 1) Thinking 1) Identify the parts. 2) What rule should I use, and why? 3) Write the rule. 4) Substitute into the proportion. 5) Solve for x. Example 2 Thinking 1) Identify the parts. 2) What rule should I use, and why? 3) Write the rule. 4) Substitute into the proportion. 5) Solve for x.

24 Put a check mark in a box that best describes how you feel you understand SAASHLLS! Practice for Success! 1. Solve for x and the length of AB. 2. In the diagram below, the length of the legs and of right triangle ABC are 6 cm and 8 cm, respectively. Altitude. is drawn to the hypotenuse of a) Solve for side AB in the right triangle ABC b) What is the length of to the nearest tenth of a centimeter? 1) 3.6 2) 6.0 3) 6.4 4) Solve for AB in simplest radical form:

25 4. The accompanying diagram shows part of the architectural plans for a structural support of a building. PLAN is a rectangle and. Which equation can be used to find the length of? 1) 3) 2) 4) Find the length of AC: 7. Four streets in a town are illustrated in the accompanying diagram. If the distance on Poplar Street from F to P is 12 miles and the distance on Maple Street from E to M is 10 miles, find the distance on Maple Street, in miles, from M to P.

26 8-4 Homework 1) In the diagram below of right triangle ACB, altitude intersects at D. If and, find the length of in simplest radical form. 2) The drawing for a right triangular roof truss, represented by, is shown in the accompanying diagram. If is a right angle, altitude meters, and is 6 meters longer than, find the length of base in meters. 3)

27 4. Two groups of students are on a camping trip they are trying to find a good place to set up camp. In the diagram below, the line of sight from Paul s Group Camp, P, to Lea s Group Camp, L, on the beach of a lake is perpendicular to the path joining the camp base, C, and the first aid station, F. The campground is 0.35 mile from where Lea set up camp. The straight paths from both the campground and first aid station to where Paul set up camp is perpendicular..45 miles.35 miles If the path from the Paul to the campground is 0.45 miles, determine and state the distance between the Paul and Lea s camp to the nearest hundredths. Paul and Lea s Supervisor, Mr. D, believes the distance from the first aid station to the campground is at least 0.5 miles. Is Mr. D correct? Justify your answer. (HINT: TRUST YOURSELF YOUR NUMBERS MAY BE DECIMALS)

28 8-5 Notes Angle Bisector Theorem Today s Goals: How can we use the angle bisector theorem to find missing segments in a given triangle? Together! What type of segment is AP? How do you know? With a partner! What do all three diagrams above have in common? Do you notice any relationships that exist?

29 ANGLE BISECTOR THEOREM: An of a vertex angle in a triangle divides the side in two segments that are to the other two sides of the triangle. LET S TRY SOME: 1. Given that is the angle bisector of, divides the sides of the triangle proportionally. This proportion can be representd as, *Is is true that AB * DC = AC * BD? Explain why. 2. Look at the measurements given below to determine if is or is not an angle bisector. Explain your reasoning.

30 Stations Practice - Work Space

31 Stations Practice - Work Space

32 8-5 Homework 1. Use the angle bisector theorem to find the missing side length of the segment in the triangle below: CA = 12, CD = 6, BA = 15, DB =? 2. What is the length of WX in the triangle below, given that WZ = 24, ZY = 12, XY = In the diagram below of, D is the midpoint of, O is the midpoint of, and G is the midpoint of. If AC = 20, AT = 36, and CT = 22, what is the perimeter of parallelogram CDOG? 1) 42 2) 50 3) 78 4) 32

33 4. Factor: 8y 3 z + 16xy 5. Solve: z 2 2z = Is AD an angle bisector? Explain why or why not. Follow up: can we identify a shortcut method that would allow us to conclude that 7. Using properties of similarity and specific vocabulary we ve learned in this unit, describe why the two triangles below are similar to each other.

34 8-6 Notes Mean Proportional and Angle bisector Theorem Practice Today s goal: How do we apply what we know about right triangle ratios and angle bisector ratios? Do now- on your own! SAASHLLS VS ANGLE BISECTOR THEOREM Examine the problems below, IDENTIFY which theorem you would use to solve for x and explain what helped you identify it. DO NOT ACTUALLY SOLVE. 1.. Solve for x 2., solve for x x 3. Solve for x 4.

35 Working backwards! Identifying SAASHLLS In shown below, altitude is drawn to at U. If,, and, which value of h will make a right triangle with as a right angle? 1) If Triangle RST were a right triangle with a right angle at and an altitude drawn to RT ( its hypotenuse), what proportions would exist? 2) We are solving for a value of h, which proportion should we use in this case? Therefore: If h is the proportion would be true, must be. SELF ASSESS FOR SUCCESS! Ask me for as hint sheet! Ask me for as hint sheet! Get to work!

36 Practice for Success! Ask me for a hints page if needed! 1) Directions: The following problem and solution comes from a student s geometry test. The student did not receive full credit for this work. Analyze the solution, by explaining in words, what the student did correctly and incorrectly. Then, solve the problem correctly. 2) The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments with lengths in the ratio 1:2. The length of the altitude is 8. How long is the hypotenuse? 3)

37 4) a) b) 5) Solve for q. 6) In simplest radical form

38 8-6 Homework Directions: Answer all of the questions on the assignment to the best of your ability. Show all work. 1. The accompanying diagram shows a 24-foot ladder leaning against a building. A steel brace extends from the ladder to the point where the building meets the ground. The brace forms a right angle with the ladder. What is the length of the steel brace in simplest radical form? How far is the bottom of the ladder from the wall? (label it y) 2. In the diagram below of right triangle ABC, altitude is drawn to hypotenuse,, and. What is the length of? 1) 2) 3) 4) 12

39 3. In triangle DEF, the altitude is drawn in to hypotenuse. Leg is 10 inches long. If is 10 inches longer than, what is the length of the hypotenuse? 4. Given AC = 42, DF = 23, AB = 48. D, E, F are midpoints. Find the perimeter of 5. If, m and m = 70, what is? 6. The ratio of the perimeters of two similar triangles is 3:7. Find the ratio of the areas and the sides. 7. Solve for the value of x.