Math  /  Geometry

Question2ABCD2 A B C D is a rhombus of center OO, having a side of 4 cm and such that A^=60\hat{A}=60^{\circ}. I,J,KI, J, K and LL are the midpoints of [AD],[AB][A D],[A B], [BC][B C] and [CD][C D] respectively. 11^{\circ} Replace the symbol * by a point from the figure : a) AIundefined=Kundefined\overrightarrow{A I}=\overrightarrow{K^{*}} b) C=KOundefined\vec{C} * *=\overrightarrow{K O} c) WDundefined=JOundefined\overrightarrow{W_{D}}=\overrightarrow{J O} d) OL=J\vec{O} L=\vec{J} e) KBundefined=BIundefined\overrightarrow{K B}=\overrightarrow{B I} f) ODundefined=Bundefined\overrightarrow{O D}=\overrightarrow{B *} 22^{\circ} Name the vectors equal to IJundefined\overrightarrow{I J} and equal to AIundefined\overrightarrow{A I}. 33^{\circ} Construct vector ARundefined=DBundefined\overrightarrow{A R}=\overrightarrow{D B} and vector BPundefined=DAundefined\overrightarrow{B P}=\overrightarrow{D A}. What do you notice ? 44^{\circ} Answer by True or False. a) AIundefined=AJundefined\overrightarrow{A I}=\overrightarrow{A J} b) DLundefined=BKundefined\overrightarrow{D L}=\overrightarrow{B K} folse c) IJundefined=LKundefined\overrightarrow{I J}=\overrightarrow{L K} Tfue d) ARundefined=CDundefined\overrightarrow{A R}=\overrightarrow{C D} falsc e) ABundefined=CBundefined\overrightarrow{A B}=-\overrightarrow{C B} f) ADundefined=CBundefined\overrightarrow{A D}=-\overrightarrow{C B} True g) BIundefined=KDundefined\overrightarrow{B I}=\overrightarrow{K D}. True 55^{\circ} Complete by == or \neq a) ADundefinedCDundefined\|\overrightarrow{A D}\| \ldots\|\overrightarrow{C D}\| b) BCundefined=ADundefined\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AD}} c) IJundefinedKLundefined\overrightarrow{I J} \leftrightarrows \overrightarrow{K L} d) BJundefinedLDundefined\overrightarrow{B J} \therefore \overrightarrow{L D} e) IKundefinedJLundefinedmagit\|\overrightarrow{I K}\| \ldots\|\overrightarrow{J L}\|-\operatorname{mag} i t ade f) OIundefined.OKundefined\overrightarrow{O I} . \ldots \overrightarrow{O K} g) LOundefinedOJundefined\overrightarrow{L O} \doteqdot \overrightarrow{O J} h) CKundefined=IAundefined\overrightarrow{C K}=\overrightarrow{I A}. 66^{\circ} Complete by the convenient vector : a) The opposite of BCundefined\overrightarrow{B C} is CRC R b) Vectors AJundefined\overrightarrow{A J} and ......... are equal. c) Vectors LDundefined\overrightarrow{L D} and JRJ R. are opposite. \qquad d) The opposite of vector ABundefined+CDundefined\overrightarrow{A B}+\overrightarrow{C D} is 77^{\circ} Write ABundefined\overrightarrow{A B} as a sum of three vectors, then of four vectors. 88^{\circ} Calculate : a) AIundefined+IBundefined\|\overrightarrow{A I}+\overrightarrow{I B}\| b) ADundefined+OJundefined\|\overrightarrow{A D}+\overrightarrow{O J}\| c) AOundefined+BJundefined\|\overrightarrow{A O}+\overrightarrow{B J}\| d) ADundefined+CBundefined\|\overrightarrow{A D}+\overrightarrow{C B}\|.

Studdy Solution

STEP 1

1. The rhombus ABCDABCD has all sides equal to 4 cm.
2. The diagonals of the rhombus bisect each other at right angles.
3. The midpoints I,J,K,LI, J, K, L divide the sides into equal segments.
4. The angle A^=60\hat{A} = 60^\circ implies the rhombus is not a square.
5. Vectors are defined in terms of their direction and magnitude.

STEP 2

1. Analyze the given vector relationships and replace the symbols.
2. Identify vectors equal to IJundefined\overrightarrow{I J} and AIundefined\overrightarrow{A I}.
3. Construct vectors ARundefined=DBundefined\overrightarrow{A R} = \overrightarrow{D B} and BPundefined=DAundefined\overrightarrow{B P} = \overrightarrow{D A}.
4. Evaluate statements as True or False.
5. Complete vector equations with == or \neq.
6. Identify convenient vectors for given conditions.
7. Express ABundefined\overrightarrow{A B} as a sum of three and four vectors.
8. Calculate magnitudes of given vector sums.

STEP 3

a) AIundefined=KLundefined\overrightarrow{A I} = \overrightarrow{K L} because both vectors represent half the diagonal of the rhombus. b) CKO=KOundefined\vec{C} K O = \overrightarrow{K O} because KOundefined\overrightarrow{K O} is the vector from KK to the center OO. c) WDundefined=JOundefined\overrightarrow{W_{D}} = \overrightarrow{J O} implies WDW_{D} is the same as OO. d) OL=J\vec{O} L = \vec{J} because both are half-diagonal vectors. e) KBundefined=BIundefined\overrightarrow{K B} = \overrightarrow{B I} because both vectors are equal halves of the side of the rhombus. f) ODundefined=BDundefined\overrightarrow{O D} = \overrightarrow{B D} because both are half-diagonals.

STEP 4

Vectors equal to IJundefined\overrightarrow{I J} are KLundefined\overrightarrow{K L} because they are parallel and equal in magnitude. Vectors equal to AIundefined\overrightarrow{A I} are CKundefined\overrightarrow{C K} because they are parallel and equal in magnitude.

STEP 5

Construct ARundefined=DBundefined\overrightarrow{A R} = \overrightarrow{D B} by drawing a vector from AA to a point RR such that it is equal in magnitude and direction to DBundefined\overrightarrow{D B}. Construct BPundefined=DAundefined\overrightarrow{B P} = \overrightarrow{D A} by drawing a vector from BB to a point PP such that it is equal in magnitude and direction to DAundefined\overrightarrow{D A}. Notice that both vectors complete the parallelogram formed by the diagonals.

STEP 6

a) False, AIundefinedAJundefined\overrightarrow{A I} \neq \overrightarrow{A J} because they are not parallel. b) False, DLundefinedBKundefined\overrightarrow{D L} \neq \overrightarrow{B K} because they are not equal. c) True, IJundefined=LKundefined\overrightarrow{I J} = \overrightarrow{L K} because they are parallel and equal. d) False, ARundefinedCDundefined\overrightarrow{A R} \neq \overrightarrow{C D} because they are not equal. e) True, ABundefined=CBundefined\overrightarrow{A B} = -\overrightarrow{C B} because they are opposite. f) True, ADundefined=CBundefined\overrightarrow{A D} = -\overrightarrow{C B} because they are opposite. g) True, BIundefined=KDundefined\overrightarrow{B I} = \overrightarrow{K D} because they are equal.

STEP 7

a) ADundefined=CDundefined\|\overrightarrow{A D}\| = \|\overrightarrow{C D}\| because all sides of a rhombus are equal. b) BCundefined=ADundefined\overrightarrow{\mathrm{BC}} = \overrightarrow{\mathrm{AD}} because they are parallel and equal. c) IJundefined=KLundefined\overrightarrow{I J} = \overrightarrow{K L} because they are parallel and equal. d) BJundefinedLDundefined\overrightarrow{B J} \neq \overrightarrow{L D} because they are not equal. e) IKundefined=JLundefined\|\overrightarrow{I K}\| = \|\overrightarrow{J L}\| because they are equal. f) OIundefined=OKundefined\overrightarrow{O I} = \overrightarrow{O K} because they are equal. g) LOundefined=OJundefined\overrightarrow{L O} = \overrightarrow{O J} because they are equal. h) CKundefined=IAundefined\overrightarrow{C K} = \overrightarrow{I A} because they are equal.

STEP 8

a) The opposite of BCundefined\overrightarrow{B C} is CBundefined\overrightarrow{C B}. b) Vectors AJundefined\overrightarrow{A J} and BKundefined\overrightarrow{B K} are equal. c) Vectors LDundefined\overrightarrow{L D} and JRundefined\overrightarrow{J R} are opposite. d) The opposite of vector ABundefined+CDundefined\overrightarrow{A B} + \overrightarrow{C D} is (ABundefined+CDundefined)-(\overrightarrow{A B} + \overrightarrow{C D}).

STEP 9

Express ABundefined\overrightarrow{A B} as a sum of three vectors: AIundefined+IJundefined+JBundefined\overrightarrow{A I} + \overrightarrow{I J} + \overrightarrow{J B}. Express ABundefined\overrightarrow{A B} as a sum of four vectors: AIundefined+IKundefined+KJundefined+JBundefined\overrightarrow{A I} + \overrightarrow{I K} + \overrightarrow{K J} + \overrightarrow{J B}.

STEP 10

a) AIundefined+IBundefined=ABundefined=4 cm\|\overrightarrow{A I} + \overrightarrow{I B}\| = \|\overrightarrow{A B}\| = 4 \text{ cm}. b) ADundefined+OJundefined=AOundefined+OJundefined=4 cm\|\overrightarrow{A D} + \overrightarrow{O J}\| = \|\overrightarrow{A O} + \overrightarrow{O J}\| = 4 \text{ cm}. c) AOundefined+BJundefined=ABundefined=4 cm\|\overrightarrow{A O} + \overrightarrow{B J}\| = \|\overrightarrow{A B}\| = 4 \text{ cm}. d) ADundefined+CBundefined=0 cm\|\overrightarrow{A D} + \overrightarrow{C B}\| = 0 \text{ cm} because they are opposite vectors.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord