Question

Please help me with 7,8,and 9. They are all related to the same circle

Answer 1

`7)\:x=30\:8)x=10\:9)\:x=8`

7) We have here intersecting chords, so we can find the measure of the arcs and angles with this formula:

`\begin{gathered} m\angle AED=(1)/(2)(mAD+mCB) \\ 5x-90=(1)/(2)(3x+x) \\ 5x-90=2x \\ 5x-90+90=2x+90 \\ 5x=2x+90 \\ 3x=90 \\ (3x)/(3)=(90)/(3) \\ x=30 \end{gathered}`

8) Focusing on the circle below that, we can do the same:

`\begin{gathered} m\angle AEC=(1)/(2)(mAC+mBD) \\ 7x+50=(1)/(2)(8x+4x+120) \\ 7x+50=6x+60 \\ 7x=6x+10 \\ 7x-6x=6x+10-6x \\ x=10 \end{gathered}`

9) And finally, we can do that since the same case presents here:

`\begin{gathered} m\angle BED=(1)/(2)(mAC+mBD) \\ 10x+40=(1)/(2)(14x-32+20x) \\ 10x+40=17x-16 \\ 10x+40-40=17x-16-40 \\ 10x=17x-56 \\ 10x-17x=17x-56-17x \\ -7x=-56 \\ (-7x)/(-7)=(-56)/(-7) \\ x=8 \end{gathered}`