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If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\) B. \(\sqrt{20}\) C. \(\sqrt{22}\) D. \(\sqrt{26}\) E. \(\sqrt{30}\)

Look at the diagram below:

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).

I am having difficulties with this problem I actually got it right but I might have been lucky I tried to find the slope which i believe the equation for this is y=3x-1... would this be helpful in finding the answer and second how did you get the legnths of each side of the triangle. I am having problems trying to find each length. Could someone please go over this problem and answer thank you

I am having difficulties with this problem I actually got it right but I might have been lucky I tried to find the slope which i believe the equation for this is y=3x-1... would this be helpful in finding the answer and second how did you get the legnths of each side of the triangle. I am having problems trying to find each length. Could someone please go over this problem and answer thank you

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\) B. \(\sqrt{20}\) C. \(\sqrt{22}\) D. \(\sqrt{26}\) E. \(\sqrt{30}\)

Look at the diagram below:

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).

Answer: B

Hi Bunuel

I did not understand how did you calculate AB & BC (highlighted above). Request you to elaborate if possible

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\) B. \(\sqrt{20}\) C. \(\sqrt{22}\) D. \(\sqrt{26}\) E. \(\sqrt{30}\)

Look at the diagram below:

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).

Answer: B

Hi Bunuel

I did not understand how did you calculate AB & BC (highlighted above). Request you to elaborate if possible

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\) B. \(\sqrt{20}\) C. \(\sqrt{22}\) D. \(\sqrt{26}\) E. \(\sqrt{30}\)

Look at the diagram below:

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).

Answer: B

Hi Bunuel

I did not understand how did you calculate AB & BC (highlighted above). Request you to elaborate if possible

I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. How to find the length of each side? Which formula/concept to apply?

I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. How to find the length of each side? Which formula/concept to apply?

Equation of a Circle is x^2+y^2= r^2 ...... the points mentioned shud satisfy this equation..we get r= sqrt(5)...diameter= 2*sqrt(5)...if 2 goes inside the sqrt sign, becomes sqrt(20)..thats my answer

Equation of a Circle is x^2+y^2= r^2 ...... the points mentioned shud satisfy this equation..we get r= sqrt(5)...diameter= 2*sqrt(5)...if 2 goes inside the sqrt sign, becomes sqrt(20)..thats my answer

x^2 + y^2 = r^2 is the equation of a circle centred at the origin. The given circle is not centred ant the origin. How did you get that \(r = \sqrt{5}\)?

P.S. The correct way is given in the solution above.
_________________

Equation of a Circle is x^2+y^2= r^2 ...... the points mentioned shud satisfy this equation..we get r= sqrt(5)...diameter= 2*sqrt(5)...if 2 goes inside the sqrt sign, becomes sqrt(20)..thats my answer

x^2 + y^2 = r^2 is the equation of a circle centred at the origin. The given circle is not centred ant the origin. How did you get that \(r = \sqrt{5}\)?

P.S. The correct way is given in the solution above.

Oops...I missed the Origin part of it. Sorry !!

Omkar Kamat When The Going Gets Tough, The Tough Gets Going !!

Great solution. However I would like to know how did you realize that finding the lengths of the sides of the triangle formed by the three points would most certainly give you a clue whether this triangle was a right triangle. Frankly, when I started this problem I felt that the only way was to choose an arbitrary point as the centre of circle and equate the distances from the centre to the three points (since they would be radii). This took me a some time. Please share your thoughts.

Can also be solved by calculating the slopes of the 2 lines from these 3 points . One comes to be 3 ie ( 5-2/2-1 = 3) and other as -1/3 ie (4-5/5-2 = -1/3) so there is a right angle and the line connecting the 2 end points will be diameter.

Isn't solving it by the equation of circle a faster and certain method?

if we hadn't calculated the distances specifically (it doesn't strike naturally to use the distance formula here), we couldn't know then that it is in fact a right angle.

can anyone explain why triangle ABC is 45-45-90 degree ? i know that one angle must be 90 degrees but the other two angles could be different and all three angles sum is 180. Are sides AB and BC similar ? If so, how ?

can anyone explain why triangle ABC is 45-45-90 degree ? i know that one angle must be 90 degrees but the other two angles could be different and all three angles sum is 180. Are sides AB and BC similar ? If so, how ?