Question

A circle's center is at `(2 ,4 )` and it passes through `(7 ,6 )`. What is the length of an arc covering `(15pi ) /8` radians on the circle?

Answer 1

Length of the arc: `31.7` (approx.)

Explanation:

If the circle has a center at `(2,4)` and passes through `(7,6)` then it has a radius of
`color(white)("XXX")r=sqrt((7-2)^2+(6-4)^2)=sqrt(25+4) =sqrt(29)`
and a diameter of
`color(white)("XXX")d=2r=2sqrt(29)`

The circumference of the circle would be
`color(white)("XXX")"Circumference"_circ=pid = 2sqrt(29)pi`

The complete circle's circumference is covered by an arc of `2pi=(16pi)/8`

An arc of `(15pi)/8` will cover `((15pi)/8)/((16pi)/8)=15/16` of `"Circumference"_circ`

The given arc will cover
`color(white)("XXX")15/16xx2sqrt(29)pi`

`color(white)("XXX")~~31.72123912` (with a calculator)