Question

The center of the circle is G. Below is a student's work to find the

value of x. Explain the error and find the correct value of x.

Answer 1

The error are

1) The use of the angle addition postulate, for
`m \widehat{ACD}`

2) The assumption that
`m \widehat{BC}` = 90°

The vertical angle property between
`m \widehat{CD}` and
`m \widehat{FA}` should be used to find 'x'

x = 5

Explanation:

From the drawing of the circle, we are given;

AD = A diameter of the circle

∴
`m \widehat{ACD}` = 180°

By angle addition postulate, we have;

`m \widehat{ACD}` =
`m \widehat{AB}` +
`m \widehat{BC}` +
`m \widehat{CD}`

By substitution property of equality, we have;

180° =
`m \widehat{AB}` +
`m \widehat{BC}` +
`m \widehat{CD}`

`m \widehat{AB}` = 5·x°

`m \widehat{BC}` = Not given

`m \widehat{CD}` = 15·x°

However;

`m \widehat{CD}` is the vertically opposite angle to
`m \widehat{FA}`

∴
`m \widehat{CD}` =
`m \widehat{FA}` = (16·x - 5)°

`m \widehat{CD}` = 15·x° = (16·x - 5)°

15·x° = (16·x - 5)°

∴ 5 = 16·x° - 15·x° = x

x = 5

`m \widehat{BC}` = 180° - (
`m \widehat{AB}` +
`m \widehat{CD}`) = 180° - (15·x° + 5·x°) = 180° - (20·x°)

`m \widehat{BC}` = 180 - (20×5)° = 80°

`m \widehat{BC}` = 80°

The error are the use of the angle addition postulate, for
`m \widehat{ACD}`, and the assumption that
`m \widehat{BC}` = 90° rather than the vertical angle property between
`m \widehat{CD}` and
`m \widehat{FA}` in trying to find 'x', from which x = 5.