## Solve Lesson 43 Page 92 SBT Math 10 – Kite>

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 $$\Rightarrow \overrightarrow {GA} + \overrightarrow {GB} + \overrightarrow {GD} = \overrightarrow 0$$