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Solve Lesson 43 Page 92 SBT Math 10 – Kite>

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Topic

For quadrilateral ABCD is a parallelogram. Call O is the intersection of two diagonals, E is the midpoint of AD, WOOD is the intersection of BEIGE and AC. Calculate:

a) \(\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} \)

b) \(\overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GD} \)

Solution method – See details

Step 1: Use the property O is the midpoint AC, BD to calculate \(\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} \)

Step 2: Prove WOOD is the centroid of the triangle ABD then calculate \(\overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GD} \)

Detailed explanation

a) Do ABCD is a parallelogram, so O is the midpoint AC and BD

\( \Rightarrow \overrightarrow {OA} + \overrightarrow {OC} = \overrightarrow 0 ,\overrightarrow {OB} + \overrightarrow {OD} = \overrightarrow 0 \) \(\begin{array}{l} \Rightarrow \ overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} = \left( {\overrightarrow {OA} + \overrightarrow {OC} } \right) + \left( {\overrightarrow { OB} + \overrightarrow {OD} } \right)\\ = \overrightarrow 0 + \overrightarrow 0 = \overrightarrow 0 \end{array}\)

b) Consider triangle ABD yes POND and BEIGE are two medians intersecting at WOOD

\( \Rightarrow \) WOOD is the focusABD \( \Rightarrow \overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GD} = \overrightarrow 0 \)

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