Homework Help

I need some help on this question please! It says... Points B and D are points of...

hardworking-human's profile pic

Posted via web

dislike 1 like

I need some help on this question please! It says...

Points B and D are points of tangency. Find the two possible lengths of AB.

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)

2 Answers | Add Yours

jeew-m's profile pic

Posted (Answer #1)

dislike 1 like

If B and D are tangents then line BC is perpendicular to AB. Similarly CD is perpendicular to AD.

Then triangle ABC and ADC become right triangles.

For both triangles AC is the hypotenuse. BC and CD are radius of the circle.

By Pythagoras theorem;

`AC^2 = AB^2+BC^2`

`AC^2 = AD^2+CD^2`

 

`AD^2+CD^2 = AB^2+BC^2`

But `BC = CD` (radius of circle)

 

`AD^2 = AB^2`

`AD = AB`

`x+10 = x^2+x-6`

`x^2 = 16`

`x = +-4`

 

When x = 4

`AB = x^2+x-6 = 16+4-6 = 14`

 

When x = -4

`AB = x^2+x-6 = (-4)^2-4-6 = 6`

 

So the possible length of AB is 14 and 6.

 

 

 

 

Sources:

Top Answer

aruv's profile pic

Posted (Answer #2)

dislike 1 like

A tangents to a circle from an exterior point are equal in length.

So

AB=AD

AB=x^2+x-6

AD=x+10

Thus

`x^2+x-6=x+10`

`x^2=10+6`

`x^2=16`

`x=+-4`

`If `

`x=4`

`AB=16+4-6=14`

`AD=4+10=14`

If  x=-4

AB=16-4-6=6

AD=-4+10=6

Thus possible length of two tangents are 14 or 6

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes