Question

Please help to find the answer to a b and c please

Answer 1

a) 5p (or 5q)

b) 5p ( or 5q)

c) 10p (or 10q)

Explanation:

We will use the following information for solving and using the given figure

• In a regular hexagon all six sides are equal
  Therefore
  
  `\overrightarrow{AB} = \overrightarrow{BC} = \overrightarrow{CD} = \overrightarrow{DE} = \overrightarrow{EF} = \overrightarrow{FA}`
  
• The circumradius R which is the length of the segment from O to the center of the hexagon to any of the vertices A, B, C, D, E, F is the exact same. So in the figure,
  
  `\overrightarrow{OA} = \overrightarrow{OB} = \overrightarrow{OC} = \overrightarrow{OD} = \overrightarrow{OE} = \overrightarrow{OF}`
  
• The distance between a pair of opposite vertices is twice the circumradius and is the same for all
  
  `\overrightarrow{EB} = 2\overrightarrow{OB}` and similarly for all other pairs

Preliminary computation

We have
`\overrightarrow{AB} = \overrightarrow{BC}`

Therefore 4p + q = 5p

5p = 4p + q

5p-4p = q

p = q

`\overrightarrow{AB} = 4p + q = 4p + p = 5p = 5q\\`

So each side is 5p in length which is also equal to 5q since p = q

Part a

`\overrightarrow{AO} = \overrightarrow{OA} = \overrightarrow{AB} = 5p` (same as 5q)

Part b

`\overrightarrow{OB} = \overrightarrow{OA} = 5p` (same as 5q)

Part c

`\overrightarrow{EB} = 2 \cdot \overrightarrow{OB} = 2 \cdot 5p = 10p` (also 10q)