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STAR CITY Math / Geometry / Midsegments Name _________________________________ Teacher ____________________ Period _____ 2. Use the figure below to answer question 1. 1. A triangular jogging path ABE is located in the park. The dimensions are EB = 92 ft., AE = 78 ft., and AB = 140 ft. The park administrator wants to make an additional path ¯ . What is the using the midsegment DC perimeter of the new path ABCD? ¯? What is the length of OR A. 155 ft. A. 2.5 ft. B. 295 ft. B. 3 ft. C. 310 ft. C. 3.5 ft. D. 380 ft. D. 10 ft. 1 STAR CITY – Math – Geometry – Midsegments Use the figure below to answer question 3. 3. Use the figure below to answer question 4. In ΔHNS , A and D are midpoints. What is the value of x? A. 8 4. B. 10 The figure above represents a light pole on a ¯ ⊥ GT ¯ and H is the midpoint of runway. If LH ¯ , which term does not identify IH̄ ? TG C. 11 A. midsegment D. 15 B. median C. altitude D. perpendicular bisector 2 STAR CITY – Math – Geometry – Midsegments 5. ¯ as a midsegment? Which triangle has BD Use the figure below to answer question 6. A. B. 6. C. D. 3 ¯? What is the length of the midsegment BE A. 4 cm B. 8 cm C. 11 cm D. 16 cm STAR CITY Open-Response / Geometry / Midsegments Name _________________________________ Teacher ____________________ Period _____ GEOMETRY OPEN-RESPONSE ITEM A A. ¯ and AF ¯ , that intersect to form an angle of 52°. The An A-frame bookshelf has two congruent supports, AE shelves divide each support into four congruent lengths. The distance between each pair of shelves is 18 ½ inches. 1. What is the measure of ∠ DCH ? Show or explain your work. 2. If the distance between supports for the middle shelf is 32 inches, find the distance between the ¯ . Show or ¯ and the distance between the supports on the floor, EF supports for the top shelf, BJ explain your work. ¯ rounded to the nearest hundredth of an inch? Show or explain your work. 3. What is the length of HG BE SURE TO LABEL YOUR RESPONSE (1), (2), AND (3). 4 STAR CITY Student Answer Document / Geometry / Midsegments Name _________________________________ Teacher ____________________ Period _____ A 5 GEOMETRY OPEN-RESPONSE ITEM A (T.2.G.3E, T.2.G.1B, T.2.G.1C) A. ¯ and AF ¯ , that intersect to form an angle of 52°. The shelves divide each An A-frame bookshelf has two congruent supports, AE support into four congruent lengths. The distance between each pair of shelves is 18 ½ inches. 1. What is the measure of ∠ DCH ? Show or explain your work. 2. If the distance between supports for the middle shelf is 32 inches, find the distance between the supports for the top shelf, ¯ and the distance between the supports on the floor, EF ¯ . Show or explain your work. BJ ¯ rounded to the nearest hundredth of an inch? Show or explain your work. 3. What is the length of HG BE SURE TO LABEL YOUR RESPONSE (1), (2), AND (3). RUBRIC FOR GEOMETRY OPEN-RESPONSE ITEM A (T.2.G.3E, T.2.G.1B, T.2.G.1C) Description Score 4 Student scores 4 points. All explanations are clear and correct. 3 Student scores 3 - 3 ½ points. 2 Student scores 2 - 2 ½ points. 1 Student scores ½ - 1 ½ points. or The response demonstrates minimal understanding of solving problems involving relationships in trapezoids and triangles. 0 Student shows no understanding of the problem. B Solution and Scoring Scoring: Part 1: Score 1 point for finding the correct measure of ∠ DCH with work shown or explanation given. OR Score ½ point for finding the correct measure of ∠ DCH with no work shown or explanation given. OR Score ½ point for correct procedure to find the measure of ∠ DCH with calculation error in work shown. Part 2: Score 2 points total: ¯ with work shown or explanation given. Score 1 point finding the correct length of BJ OR ¯ with no work shown or explanation given. Score ½ point for finding the correct measure of BJ OR ¯ with calculation error in work shown. Score ½ point for correct procedure to find the measure of BJ AND ¯ with work shown or explanation given. Score 1 point finding the correct length of EF OR ¯ with no work shown or explanation given. Score ½ point for finding the correct measure of EF OR ¯ with calculation error in work shown. Score ½ point for correct procedure to find the measure of EF ¯ with work shown or explanation given. Part 3: Score 1 point finding the correct length of HG OR ¯ with no work shown or explanation given. Score ½ point for finding the correct measure of HG OR ¯ with calculation error in work shown. Score ½ point for correct procedure to find the measure of HG Note: In Part 2 and Part 3, including units is a 3 / 4 issue only. To receive a score of 4, the student must include correct units, but there is no deduction for missing or incorrect units below a score of 4. Sample Solution 1: Part 1: 1 2 (180o − 52o) = 64o 1 2 ∠ ABJ = 64o = ∠ CDG ∠ AJB = 64o = ∠ HGD ∠ ABJ = 64o OR corresponding angles are congruent 1 o ( 2 360 ∠ BCH + ∠ DCH = 180olinear pairs − 2(64o)) = 116o ΔABJ is similar to ΔACH by ∠ ABJ = ∠ BCH corresponding angles are congruent ∠ DCH = 180o − 64o = 116o ∠ DCH = 116o Part 2: (180o − 52o) = 64o OR ¯ is the midsegment of ΔACH . BJ AA similarity ¯= BJ 1 ( ) 2 CH ¯= BJ 1 ( ) 2 32 ¯ = 16 inches BJ ΔAEF is 4 times greater than ΔABJ . ¯ ) = 4(16) = 64 inches. 4( BJ Part 3: ( AJ ¯ )2 = 2 1 2 ¯ ) + (18 2 ) ( 12 BJ ( 1 ( AJ ¯ )2 = (8)2 + 18 2 )2 ( AJ ¯)2 = 406.25 ¯ = 20.16inches AJ ¯ = HG ¯ AJ ¯ = 20.16inches HG Sample Solution 2: Part 1: 1 2 (180o − 52o) = 64o 1 2 ∠ ABJ = 64o = ∠ CDG ∠ ABJ = 64o ∠ AJB = 64o = ∠ HGD OR corresponding angles are congruent 1 o ( 2 360 (180o − 52o) = 64o ∠ ABJ = ∠ BCH corresponding angles are congruent ∠ BCH + ∠ DCH = 180olinear pairs − 2(64o)) = 116o ∠ DCH = 180o − 64o = 116o ∠ DCH = 116o Part 2: Using proportions with Triangle ABJ and Triangle congruent from Part 1), ACH (because we know the angles are 1 Height ofΔABJ BJ (Base ofΔABJ ) = 18 2 Height ofΔACH , CH (Base ofΔACH ) x ¯ as the median of trapezoid Using CH BJGD, = 37 32 ¯ = 16 in. x = 16 in. so BJ ¯ as the median of trapezoid Now use DG CHFE, then ¯= CH 32 = Part 3: ( AJ ¯ )2 = 1 (¯ 2 BJ 1 ( 2 16 ¯) + DG ¯) + DG ¯= DG 48 = 1 (¯ 2 CH 1 ( 2 32 ¯) + EF ¯) + EF ¯) 32 × 2 = (16 + DG ¯) 48 × 2 = (32 + EF ¯) 64 = (16 + DG ¯) 96 = (32 + EF ¯ 64 − 16 = DG ¯ 96 − 32 = EF ¯ = 48in. DG ¯ = 64in. EF 2 1 2 ¯ ) + (18 2 ) ( 12 BJ ( 1 ( AJ ¯ )2 = (8)2 + 18 2 ( AJ ¯)2 = 406.25 ¯ = 20.16inches AJ ¯ = HG ¯ AJ ¯ = 20.16inches HG )2 STAR CITY Math / Geometry / Midsegments Answer Key and Alignment Answer 1. A 2. B 3. C 4. A 5. B 6. A Source TLI TLI TLI TLI TLI TLI SLE T.2.G.3e T.2.G.3e T.2.G.3e T.2.G.3e T.2.G.3e T.2.G.3e