Problem Solving

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}

Let point A (1,2), point B (2,5), point C (5,4)
Plot the above points roughly in the x,y axis to get an idea how the triangle would look like.

Now, if angle ABC =90degree, then AC would be the diameter of the circle, so find the distance between the points:
AB=sqrt(10)
BC=sqrt(10)
AC=sqrt(20)

So, by Pythagoras theorem if AB^2 + BC^2 = AC^2 then its a right angle triangle with angle ABC =90 degree
substituting the values in the equation you get AC=sqrt(20), which matches the distance between the point A & C and thus is the ans too...

B

how did you get sqrt(10)?

distance between two lines = Sqrt ((x2-x1)^2+(y2-y1)^2)

perfect thanks! one last thing.. how were we able to know that it was a 90 degree angle?

money9111 wrote:perfect thanks! one last thing.. how were we able to know that it was a 90 degree angle?

Using Pythagoras theorem.


In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

Ie if a, b, c were sides of a Right angled triangle then a^2+b^2 = c^2


The converse of which is also true

If a^2+b^2 = c^2 then it is a right angled triangle.

so because of the lengths which we were able to find using the other formula... we use Pyth. Th. to recognize that it's a right triangle?

money9111 wrote:so because of the lengths which we were able to find using the other formula... we use Pyth. Th. to recognize that it's a right triangle?

Spot on!

That is exactly how we find out whether it is a right angled triangle

::sigh of relief:: thanks for your enthusiasm!

money9111 wrote:so because of the lengths which we were able to find using the other formula... we use Pyth. Th. to recognize that it's a right triangle?

Alternatively, we can find slope between points (5,4) and (2,5) and compare it with slope for line between (2,5) and (1,2). For perpendicular lines, the slope must negative reciprocal of each other.
The slopes for the above lines are -1/3 and 3, which make them perpendicular to each other and thus making line between points (5,4) and (1,2) the diameter.

Thank you amittilak I think that rule will be much easier to remember... adding that to notecards right now!

money9111 wrote:Thank you amittilak I think that rule will be much easier to remember... adding that to notecards right now!

you are welcome...
I forgot to mention something in my last post. Maybe you already know this but if the slopes of the lines are equal, the lines are parallel.

i hope that i would have remembered that but thanks for the reminder! now i have no excuse to forget it!