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In the diagram below, chords \(\overline{AB}\) and \(\overline{CD}\) are perpendicular, and meet at X. If AX = 3, BX = 4, CX = 6, and DX = 2, then find the diameter of the circle.

 May 26, 2020

Best Answer 

 #4
avatar+816 
+3

Line segment EF is parallel to the line segment CD. Both line segments have the same lengths ( 8 units ). They are 1 unit apart from each other.

And finally, these line segments create an inscribed rectangle with side lengths of 8 and 1 unit. A diagonal of this rectangle is the diameter of a circle.

 CD = 8          CF = 1                            

 

 DF = sqrt ( CD² + CF² )    wink

 

 May 28, 2020
edited by Dragan  Jul 7, 2020
 #1
avatar
-3

I think you can use power of a point.

 May 26, 2020
 #2
avatar+25247 
+2

In the diagram below, chords \(\overline{AB}\) and \(\overline{CD}\) are perpendicular, and meet at X.
If AX = 3, BX = 4, CX = 6, and DX = 2, then find the diameter of the circle.

\(\text{Let the center of the circle $C(x_c,\ y_c)$ }\)

\(\begin{array}{|rcll|} \hline 4+y_c &=& 3-y_c \\ 2y_c &=& -1 \\ \mathbf{y_c} &=& \mathbf{-\dfrac{1}{2}} \\ \hline \end{array} \begin{array}{|rcll|} \hline 6+x_c &=& 2-x_c \\ 2x_c &=& -4 \\ \mathbf{x_c} &=& \mathbf{-2} \\ \hline \end{array}\)

 

\(\begin{array}{|lrcll|} \hline & (x-x_c)^2 + (y-y_c)^2 &=& r^2 \\ P(2,0): & (2-x_c)^2 + (0-y_c)^2 &=& r^2 \quad | \quad x_c = -2,\ y_c =-\dfrac{1}{2} \\ & (2+2)^2 + \left(0+\dfrac{1}{2} \right)^2 &=& r^2 \\ & 4^2 + \dfrac{1}{4} &=& r^2 \\ & 16 + \dfrac{1}{4} &=& r^2 \quad | \quad *4 \\ & 64+1 &=& 4r^2 \\ & 65 &=& 4r^2 \quad | \quad \text{sqrt both sides} \\ & \sqrt{65} &=& 2r \\ \hline \end{array}\)

 

The diameter of the circle is \(\mathbf{\sqrt{65}}\)

 

laugh

 May 27, 2020
 #3
avatar+109729 
0

Thanks Heureka, 

here is another coordinate geometry approach.

 

I just found the midpoint of the horizontal line x=-2 and drew a vertical axis of symmetry through it.

then the midpoint of the vertical points y=-0.5 and drew a horizontal axis of symmetry through it.

 

Where these axes of symmetry meet is the centre.  (-2,-0.5)

 

Now just use the distance formula between the centre and any point on the circumference to find the radius.

Then do what you need to do to find the diameter.

 

 

 

 

 May 27, 2020
 #4
avatar+816 
+3
Best Answer

Line segment EF is parallel to the line segment CD. Both line segments have the same lengths ( 8 units ). They are 1 unit apart from each other.

And finally, these line segments create an inscribed rectangle with side lengths of 8 and 1 unit. A diagonal of this rectangle is the diameter of a circle.

 CD = 8          CF = 1                            

 

 DF = sqrt ( CD² + CF² )    wink

 

Dragan May 28, 2020
edited by Dragan  Jul 7, 2020
 #5
avatar+109729 
0

That is an excellent solution Dragan,  thanks for explaining it :)

 May 28, 2020
 #6
avatar+816 
+1

urw, Melody!!! smiley

Dragan  May 28, 2020
 #7
avatar+140 
+1

Thank you guys for you help! I appreciate it

 May 29, 2020

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