Find the radius of circle. ............... #maths #geometry #olympiad #cds #cat #518 #cbse #ssc #cgl

Added: 2026-05-16

[00:00:00][clears throat] >> Welcome everyone. This interesting question, can you find the radius of this circle which is inscribed in a square?

[00:00:08]Given is the length of this red line segment is 9 cm and length of this green line segment is 2 cm.

[00:00:18]Let's check this interesting solution.

[00:00:20]Let's start by assuming the radius of circle as r cm. Thus OA length will be r and OC length will be r.

[00:00:28]We know that in a circle, radius is perpendicular to the tangent. So angle E is 90. Same logic, angle C is 90°.

[00:00:37]Now let's drop a perpendicular from point A on OC.

[00:00:42]Now in this right-angle triangle, if we can find this length AF and FO in terms of r, then we can apply Pythagoras theorem to solve it.

[00:00:51]Now if you look at this figure ABCF, all the angles are 90, so it's a rectangle. And thus SC length is same as >> [music] >> 2 cm.

[00:01:02]How about OF length? That will be full length r minus 2.

[00:01:07]Now to get AF length, let's consider this figure OEDC.

[00:01:12]It's a rectangle.

[00:01:13]So OE is r, thus CD length will also be r.

[00:01:18]Now this complete length BD, that's given as 9. So BC length will be 9 minus r.

[00:01:24]And AF length, that will be same 9 minus r.

[00:01:28]Now solving this triangle by Pythagoras theorem, we'll get r minus 2 squared plus 9 minus r squared.

[00:01:35]This is equals to hypotenuse squared r, r squared.

[00:01:38]Opening the bracket, applying A minus B whole squared identity, we'll get this as 81 plus r squared minus 18 r.

[00:01:47]And this term will be r squared plus 4 minus 4 r. This is equals to r squared.

[00:01:53]Solving it, both the side r squared will get cancelled out.

[00:01:56]Simplifying it, we'll get a quadratic in terms of r.

[00:02:00]Solving it by splitting the middle term, so -22 we'll write it down as -17 and -5.

[00:02:08]Now, taking r common and factorizing it, we'll get this as r -17 * r -5 = 0.

[00:02:15]So, the radius of circle, it can be 5 or it can be 17.

[00:02:19]But, if you look carefully, if BD length is 9 cm, >> [music] >> then the radius has to be less than 9.

[00:02:25]So, ignoring this higher value, we get radius of the circle as 5 cm and that's our answer. I hope you enjoyed the solution. I will see you in next video.

[00:02:36]Till then, ta-ta, bye-bye.