Question

In the diagram shown points A, B, C, D, and E lie on circle M. Point P is on the exterior or circle M and secants PCD and PBE drawn. It is known that m

(a) Determine the measure of BC


(b) Determine the measure of

Answer 1

The measure of
`m \hat {BC}=34^0`

The measure of
`\angle DFE = 81^0`

Explanation:

The correct question is added in the diagram below.

From the diagram;

Given that:

∠A = 55°

∠P =38°

`m \hat {AC} = 86^0`

`m \hat{DE} = 2( \angle A )`

`m \hat{DE} = 2( 55^0 )`

`m \hat{DE} = 110^0`

a) it is clear and obvious that ∠P is formed at the exterior of the circle; so:

`m \angle P = \frac{m \hat {DE} -m \hat {BC} }{2}`

`38= \frac{110 -m \hat {BC} }{2}`

`38*2={110 -m \hat {BC} }{`

`76={110 -m \hat {BC} }{`

`m \hat {BC} }{={110 -76`

`m \hat {BC}=34^0`

b)
`m \hat {AC}=m \hat {AB}+ m \hat {BC}`

`86=m \hat {AB}+ 34`

`m \hat {AB}=86-34`

`m \hat {AB}=52`

∴
`\angle DFE = \frac{m \hat{DE}+m \hat{AB}}{2}` (rule: chord intersecting inside the circle)

`\angle DFE = (110+52)/(2)`

`\angle DFE = (162)/(2)`

`\angle DFE = 81^0`