OPEN RESPONSE QUESTION FROM MCAS FALL RETEST 2003, GRADE 10, #17 INTRODUCTION OPEN RESPONSE QUESTION FROM MCAS FALL RETEST 2003, GRADE 10, #17 The following is an “Open Response” question for you to practice answering these types of questions in preparation for the State Mandated MCAS. This exercise will help to walk you through the process. There are also some reference links to give you some extra help. Next

HOW TO ANSWER AN OPEN RESPONSE QUESTION Be sure to Read all parts of each question carefully. Make each response as clear, complete and accurate as you can. Check all your work! Next

TOPIC Apply similarity correspondences (ex. ΔABC ~ ΔXYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification. Next

QUESTION #17 To measure the width of a stream indirectly, Claude placed four stakes in the ground at points B, C, D, and E. He used a rock on the opposite bank to determine point A. Triangles ABC and ADE are formed, as shown in the diagram below. Next

QUESTION #17 Next

Part A. Explain how you can show BCA is congruent to  DEA. Hint: Separate the diagram into two separate triangles. Additional Hint Next

Part B. Explain how you know BCA is similar to DEA. Hint Next

Part C. Write a proportion or an equation that can be used to determine the distance (indicated by d in the diagram) across the stream. Hint Next

Part D. What is the distance across the stream Part D. What is the distance across the stream? Show or explain how you obtained your answer. Hint Next

REFERENCES If you are having any trouble with this problem there are some links below that will help you with the standards involved with this question: http://mathforum.org/dr.math/ http://www.algebra.com/algebra/homework/coordinate/ http://school.discovery.com/homeworkhelp/webmath/ http://www.doe.mass.edu/mcas/ Next

EVALUATION This is the rubric that the state uses to rate responses to question 17. How did you do?   4 The student response demonstrates an exemplary understanding of the Geometry concepts involved in working with similar triangles calculating lengths of sides through proportions. 3 The student response demonstrates a good understanding of the Geometry concepts involved in working with similar triangles calculating lengths of sides through proportions. Although there is significant evidence that the student was able to recognize and apply the concepts involved, some aspect of the response is flawed. As a result the response merits 3 points. 2 The student response contains fair evidence of an understanding of the Geometry concepts involved in working with similar triangles calculating lengths of sides through proportions. While some aspects of the task are completed correctly, others are not. The mixed evidence provided by the student merits 2 points. 1 The student response contains only minimal evidence of an understanding of the Geometry concepts involved in working with similar triangles calculating lengths of sides through proportions. The student response contains insufficient evidence of an understanding of the Geometry concepts involved in working with similar triangles calculating lengths of sides through proportions.   Next

SOLUTIONS Solution for A Solution for B Solution for C Solution for D Next

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Additional Hint for Part A B E 32 ft 28 ft d + 6 ft d C D When you separate the two triangles, notice that  B and  D are congruent and that  A is the same angle in each of the two triangles. If you know that two angles of a triangle are congruent to two angles of another triangle, what do you know about the third angle? Go Back

Hint for Part B Two triangles are similar if all corresponding angles are congruent and all corresponding sides are in proportion. In order to prove triangles are similar you must prove one of the following similarities: SAS Similarity SSS Similarity AA Similarity Back to Part B

SAS Similarity SAS Similarity – Prove that two corresponding sides are in proportion and that the corresponding included angles, the angles located in between the sides, are congruent. Ex. Since and B = D = 90 º then ABC ~ FDE by SAS Similarity. B E 14 ft 7 ft 20 ft 10 ft C D A F 10 20 7 14 = Back to Hint Page

SSS Similarity SSS Similarity – Prove that all corresponding sides are in proportion. F A Ex. Since then ABC ~ FDE by SSS similarity. 10 ft 8 ft 25 ft 20 ft 8 20 10 25 = 6 15 = C B 6 ft E D 15 ft Back to Hint Page

AA Similarity A = F = 65º B = D = 90º AA Similarity – Prove that two corresponding angles are congruent. C B A 65º E D F Ex. Since A = F = 65º B = D = 90º then  ABC ~  FDE by AA Similarity. Back to Hint Page

Hint for Part C Once you know the triangles are similar, you can set up a proportion using the sides lengths that are given. In this problem, use C B A E D A AB AD BC DE = Back to Part C

Hint for Part D Solve the proportion you wrote in part C by cross multiplying: means that When you get your solution for d, you will have found the distance across the lake. AB AD BC DE = d d+6 28 32 = Back to Part D

Solution A ABC and  DEA are congruent because they are both right angles. Angle  BAC and  DAC are congruent because they are the same angle, reflexive property. There is a theorem that states if two angles of one triangle are congruent to two angles of another triangle, then the third angle is congruent. As an alternate solution, you could use the sum of the triangles is 180 to explain why the two angles are congruent. Back to Solution Page

Solution B Since from part A you know that two angles of one triangle are congruent to two angles of another triangle, then ABE ~ ADE by the AA Similarity Postulate. Back to Solution Page

Solution C Since you know that: Plug in the values and write the proportion: AB AD BC DE = d d+6 28 32 = Back to Solution Page

Solution D From part C: After cross-multiplying: 32d = 28(d+6) So the distance across the stream, AB, is 42 feet. d d+6 28 32 = Back to Solution Page I’m done!