NAME COMMON CORE GEOMETRY Module 2 Part II

NAMECOMMON CORE GEOMETRYModule 2 Part IISetting up proportions with similar trianglesAnd Simplifying RadicalsDATE12/11PAGE2-3HOMEWORKHomework Worksheet20-22TOPICLesson 1: Similar triangles and proportionsStart Lesson 2LESSON 2: Applying the Triangle Side SplitterTheoremLESSON 3: Applying the triangle side splittertheorem continuedLesson 4: Ratios of Sides, Perimeters, and AreasQUIZLesson 5: The Angle Bisector TheoremLesson 6: Special Relationships Within RightTriangles—Dividing into Two Similar SubTrianglesLesson 7: Special Relationships Within RightTriangles- Another useful proportionQUIZLESSON 8: A side note- what to do if we get aradical?Lesson 9: More operations with -1512/1916-1712/2218-1912/231/5/1523-24LESSON 10: Adding and Subtracting RadicalsHomework Worksheet1/625-26LESSON 11: Putting It all TogetherHomework WorksheetQUIZReviewReviewTestFinish Review Packet1/71/81/9Homework WorksheetHomework WorksheetHomework WorksheetHomework WorksheetHomework WorksheetHomework WorksheetNo HomeworkNo HomeworkStudyNo Homework1

Lesson 1: Similar triangles and proportionsProve triangles are similar:a.) In the diagram below DE is parallel to AB, mark your picture accordingly:CDEABb.) Fill in the appropriate givens and what you are trying to proveGiven:Prove:c.) Proof:d.) Draw the similar triangles separately and label:D.)Now that we know that the triangles are similar, Let’s fill in the following proportions:2

EXERCISES: In 1-3 find x given that ̅̅̅̅̅̅̅̅1.2.3.4. A vertical pole, 15 feet high, casts a shadow 12 feet long. At the same time, a nearby tree casts a shadow 40feet long. What is the height of the tree?5. Caterina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feetabove the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of thecliff with the water at the back of her boat. Her eye level is 5.5 feet above the ground. How far out from thecliff, to the nearest tenth, is Catarina’s boat?5.5ft10ft35ft3

LESSON 2: Applying the Triangle Side Splitter TheoremSide Splitter Theorem:A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.Using this Theorem, answer the following questions:1. If ̅̅̅̅̅̅̅̅ , ̅̅̅̅, ̅̅̅̅2. If ̅̅̅̅̅̅̅̅ , ̅̅̅̅, ̅̅̅̅, and ̅̅̅̅, and ̅̅̅̅, what is ̅̅̅̅ ?, what is ̅̅̅̅ ?4

4.5.6. In the diagram pictured, a large flag pole stands outside of an office building. Josh realizes that when helooks up from the ground, 60m away from the flagpole, that the top of the flagpole and the top of the buildingline up. If the flagpole is 35m tall, and Josh is 170m from the building, how tall is the building to the nearesttenth?5

LESSON 3: Applying the triangle side splitter theorem continuedOpening ExerciseIn the two triangles pictured below, ̅̅̅̅̅̅̅̅ and ̅̅̅̅̅ . Find the measure of x in both trianglesWhat is the relationship between the triangles ABC and FGH?Since the two triangles share a common side, look what happens when we push them together:Now we have three parallel lines cut by transversals, is the transversal on the left in proportion to thetransversal on the right?6

THEOREM: If 3 or more lines are cut by 2 transversals, then the segments of the transversals are inproportion.ExercisesIn exercises 1 and 2, find the value of x. Lines that appear to be parallel are in fact parallel1.)2.)3.) In the diagram below, ̅̅̅̅ ̅̅̅̅ ̅̅̅̅, AB 20, CD 8, FD 12 and AE:EC 1:3. If the perimeter of thetrapezoid ABCD is 64, find AE and EC.7

Lesson 4: Ratios of Sides, Perimeters, and AreasExerciseGiven ABC A’B’C’ pictured to the right:a. Find the lengths of the missing sides.b. Find the perimeters of the triangles.c. Find the areas of the triangles.Area formula:d. What is the ratio of the sides of the triangles?e. What is the ratio of the perimeters of the triangles?f. What is the ratio of the areas of the triangles?RULES: The ratio of the perimeters is The ratio of the areas is8

Example 2Let’s test the hypothesis we made in Exercise 1.Given the similar triangles pictured to the right, find:a.) The ratio of the sidesb.) The ratio of the perimetersc.) The ratio of the areasGiven the similar rectangles pictured below, find:a.) The ratio of the sidesb.) The ratio of the perimetersc.) The ratio of the areas9

Examples:1.) When two figures are similar and the ratio of their sides is a:b, then:The ratio of their perimeters is:The ratio of their areas is:2.) Two triangles are similar. The sides of the smaller triangle are 6,4,8. If the shortest side of the largertriangle is 6, find the length of the longest side.3.) The sides of a triangle are 8, 5, and 7. If the longest side of a similar triangle measures 24, find theperimeter of the larger triangle.4.) Find the ratio of the lengths of a pair of corresponding sides in two similar polygons if the ratio of theareas is 4:25.10

Lesson 5: The Angle Bisector TheoremThe angle bisector theorem states:In ABC, if the angle bisector of Ameets side BC at point D, then𝐵𝐷𝐶𝐷𝐵𝐴𝐶𝐴Exercises:1.) In ABC pictured below, AD is the angle bisector of A. If CD 6, CA 8 and AB 12, find BD.2.) In ABC pictured below, AD is the angle bisector of A. If CD 9. CA 12 and AB 16. Find BD.11

3.) The sides of ABC pictured below are 10.5, 16.5 and 9. An angle bisector meets the side length of 9. Findthe lengths of x and y.12

Lesson 6: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-TrianglesOpening ExerciseUse the diagram to complete parts (a)–(c).a. Are the triangles shown similar? Explain.b. Determine the unknown lengths of the triangles using Pythagorean Theorem.Example 1In ABC pictured to the right, B is a right angle and BC is the altitude.DEFINE: Altitude:a. How many triangles do you see in the figure? Draw them:the altitude ̅̅̅̅ divides the right triangle into two sub-triangles,b. InSinceIs?Is?andcan we conclude that13and? Explain.

Example 2Consider the right trianglebelow.84a. Altitude ̅̅̅̅ is drawn from vertexand the segment ̅̅̅̅ asxto the line containing ̅̅̅̅ Segment ̅̅̅̅, segment ̅̅̅̅,b. Draw the two similar triangles ADB and BDC and label the sides appropriately.c.Set a proportion to find the value ofd. We should notice that the altitude of the whole triangle is the long leg in the small triangle and is theshort leg in the medium triangle. Therefore we can really use a short cut when we see a diagram likethis:14

EXAMPLES:1.)3.) Given triangle2.)with altitude ̅ , find16,.3615

Lesson 7: Special Relationships Within Right Triangles- Another useful proportionConsider the right trianglebelow (same as we worked with yesterday).x420a. Altitude ̅̅̅̅ is drawn from vertexand the segment ̅̅̅̅ asto the line containing ̅̅̅̅ Segment ̅̅̅̅, segment ̅̅̅̅,b. Draw the two similar triangles ADB and ABC and label the sides appropriately.c.Set a proportion to find the value ofd. We should notice that the hypotenuse of the small triangle is the leg in the large triangle and theyhypotenuse in the large triangle is the leg in the short triangle. Therefore we can really use a short cutwhen we see a diagram like this:16

EXAMPLES:1.)2.)what is the length of?What is the length of3.) In the diagram below, the length of the legsandrespectively. Altitudeis drawn to the hypotenuse ofa centimeter?17?of right triangle ABC are 6 cm and 8 cm,. What is the length ofto the nearest tenth of

LESSON 8: A side note- what to do if we get a radical?Opening ExerciseSolve for x:x68Consider our answer, what can we do with it? If we round we are not using the most accurate answerso we want to leave it in simplest radical form. (we’ll come back and do this later)Perfect Squares-Identify some perfect squares below:Factors:Perfect squareFactors:12x x22x2 x233X3 x342X4 x452X5 x562X6 x672X7 x7Perfect square8292*make a note about this:10211218

Write the square roots of the following:1.2. 4. 3. 5. Simplifying Non-Perfect squares: 𝑛𝑜𝑛 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑟𝑒 𝐿𝑎𝑟𝑔𝑒𝑠𝑡 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟Simplify:6. 7. 8. 9. 10. 11.) go back and simplifyyour answer from theopening exercise.19

Lesson 9: More operations with radicals1.Complete parts (a) through (c).a. Compare the value of to the value of b. Make a conjecture about the validity of the following statement. For nonnegative real numbers, . Explain.c.2.Does your conjecture hold true forandand?Complete parts (a) through (c).a. Compare the value of to the value of .b. Make a conjecture about the validity of the following statement. For nonnegative real numbers, whenc., . Explain.Does your conjecture hold true forand20?and

Exercises 3–12Simplify each expression as much as possible and rationalize denominators when applicable.3. 5. 7. 9. 10. 12.11. 4. 6. 8.21

13.Find the area of the figure below:14.Calculate the area of the triangle:22

LESSON 10: Adding and Subtracting Radicals1.)a. Calculate the perimeter of the triangle below:b. Calculate the perimeter of the triangle: Since the radicals are not the same we need to do some work before we can add.*Simplify each side then add.23

SIMPLIFY2.) 4.) .3.) 5.)Find the Perimeters6.)7.)24

LESSON 11: Putting It all TogetherOpening ExerciseIn the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse.Recall the proportions that we set up back in lessons 6 and 7 and fill in the appropriate proportions below:𝒙𝒄𝒄𝒛𝒃𝒂1.) Triangle ABC shown below is a right triangle with altitudeIfanddrawn to the hypotenuse.Write all answers in simplest radical form:a.) what is the length of AD?C.) what is the length of AC?b.) What is the length of AB?D.) What is the area of triangle ABC?25.

2.) Given the diagram below, find:416a.) Length of ABb.) Length of ADc.) Length of ACD.) Perimeter of triangle ABC.E.) Express the area of triangle ABC in simplest form:26

Lesson 5: The Angle Bisector Theorem The angle bisector theorem states: Exercises: 1.) In ABC pictured below, AD is the angle bisector of A. If CD 6, CA 8 and AB 12, find BD. 2.) In ABC pictured below, AD is the angle bisector of A. If CD 9. CA 12 and AB 16. Find BD. In ABC, if the angle bisector of A meets side BC at point D, then