Question

Study the diagram of circle G, where two chords, EF and WV, are congruent.Segment JK is a diameter of circle G and intersects EF at a right angle at point PJK intersects WV at a right angle at point O.

Answer 1

Since EF and WV are congruent, and the intersection of EF and WV to the diameter forms two right angles, they are parallel. We can conclude therefore that PG is congruent with GO

We can say that

`\begin{gathered} \overline{PG}=\overline{GO} \\ \\ \text{Substitute the following given} \\ \overline{PG}=x-4 \\ \overline{GO}=(1)/(2)x+3 \\ \\ \overline{PG}=\overline{GO} \\ x-4=(1)/(2)x+3 \\ \\ \text{Subtract both sides by }(1)/(2)x,\text{ and also add both sides with }4 \\ x-4-(1)/(2)x+4=(1)/(2)x+3-(1)/(2)x+4 \\ x-(1)/(2)x-4+4=(1)/(2)x-(1)/(2)x+3+4 \\ x-(1)/(2)x\cancel{-4+4}=\cancel{(1)/(2)x-(1)/(2)x}+3+4 \\ (1)/(2)x=7 \\ \\ \text{Multiply both sides by }2,\text{ to get rid of the coefficient in the left side} \\ 2\mleft((1)/(2)x=7\mright)2 \\ x=14 \end{gathered}`

Now that we have solved for x, substitute it to the given of PG

`\begin{gathered} \overline{PG}=x-4 \\ \overline{PG}=14-4 \\ \overline{PG}=10 \end{gathered}`

Therefore, PG is equal to 10 units.