On part of this worksheet we got a extra credit question. There is a triangle within a circle.http://i47.tinypic.com/209od3o.jpg this is what i'm talking about. The question is to find the length...

On part of this worksheet we got a extra credit question. There is a triangle within a circle.

http://i47.tinypic.com/209od3o.jpg 

this is what i'm talking about. The question is to find the length of the diameter. How is this possible/how do you do it? Help would be appreciated

Triangle is inscribed with 6, 10, 8 of sides. 

Asked on by zigul

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jeew-m's profile pic

jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted on

Let AB is the diameter of the circle and O is the center as seen in your figure.

`AB = d`

Let the following points are in such a way that;

`AC = 8`

`CD = 6`

`AD = 10`

 

By inscribed angle therem;

2`angleBDA` = `angleAOB`

But `angleAOB = 180 deg.`

 

Therefore `angleBDA = 90 deg`

 

So BDA is a right triangle.

Using Pythagoras's law;

`AB^2 = BD^2+AD^2`

`d^2 = BD^2+10^2`

 

But BCD is also a right triangle.

`BD^2 = BC^2+CD^2`

`BD^2 = (d-8)^2+6^2`

 

`d^2 = BD^2+10^2`

`d^2 = (d-8)^2+36+100`

`d^2 = d^2-16d+64+36+100`

`16d = 200`

`d = 200/16`

`d = 12.5`

 

So the diameter is 12.5

 

 

Sources:
oldnick's profile pic

oldnick | (Level 1) Valedictorian

Posted on

the triangle is rectangle, indeed:

`6^2+8^2=10^2`   with hypotenuse of 10.

Now for a geometrical theorem any point of a circle linked with the ends of the diameter has an angle of `pi/2` ,so the diameter is the hypotenuse of the triangle. 

 

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