A Nice Geometry Problem – Only 1% Can Solve This Geometry Problem?

Added: 2026-05-12

[00:00:00]Hello everyone, you are welcome.

[00:00:03]Today we have a very interesting geometric question.

[00:00:07]Here we have given a circle said that there are given three line segments which are perpendicular to each other.

[00:00:14]The smaller has a length of two units and the other two has the same length, four units.

[00:00:19]And we have also given that AB is the diameter of this circle.

[00:00:24]Here our target is to find out the radius of the circle.

[00:00:28]To find out the radius of this circle here first we will try to expand this one line segment such that this line means segment join the point on the circle.

[00:00:37]And then we'll connect that point to point A and we will also draw diameter of this circle.

[00:00:44]So this figure will become Now after expanding this perpendicular and joining this point with point A here this angle is a right angle. So this angle will be also a right angle. This is because the angle on a straight line is always 180°.

[00:01:00]Now we'll try to prove this one angle that is this angle is a right angle or not. Here we will try to prove this angle is a right angle.

[00:01:08]So find it here we will draw the diameter of this circle first.

[00:01:12]Now to prove this angle is a right angle here we will use a result.

[00:01:17]We know that if we have a triangle inside the semicircle said that the bigger side or the hypotenuse of the triangle is the diameter of the circle and the triangle in the semicircle will be a right right angle triangle because the angle to the front of the hypotenuse that will be a right angle. So therefore here this angle is a right angle as well.

[00:01:40]Now look to this one close figure. Let us suppose this is point C.

[00:01:44]And in this figure these three angles are right angles, 90° angles. So therefore the third angle will also the right angle because here the sum of all the angles in a quadrilateral or rectangle is 360°.

[00:01:59]So, therefore, here this figure is a rectangle having same opposite sides.

[00:02:04]So, this side is two, so this side will be also two.

[00:02:08]And this side is four, so this side will be also four.

[00:02:12]And here we have a right angle triangle, triangle ABC.

[00:02:16]So, here in this right angle triangle, we will apply in the Pythagoras theorem, and we will find out the radius of this circle. So, by Pythagoras First, we will try to find out the length of AB, that is the diameter of this circle. And then we'll divide the diameter by two, we will get the radius.

[00:02:33]So, here by Pythagoras theorem, the square of hypotenuse is equal to perpendicular square plus base square.

[00:02:42]So, therefore, from this figure here, we can write by Pythagoras theorem here, AB whole square that is equal to AC square plus BC square.

[00:03:03]So, substitute the values from the figure. So, here the value of AB, it is not given. The value of AC is four units, and the value of BC, which is 4 + 2 units, means 6 units.

[00:03:15]So, we substitute these values here, so this equation will become this AB whole square that will be the same.

[00:03:24]And here the value of AC, it is simply four, so this will become four square plus value of BC is six, so this will become six square.

[00:03:32]And this right hand side will become here four square is simply it is 16.

[00:03:37]Plus and six square is 36.

[00:03:41]Let us sum up these two values, so therefore, the value of A * B whole square that will become 6 + 6 it is 12.

[00:03:50]3 + 1 is 4, 4 + 1 is 5. This is 52.

[00:03:54]And we will take square root on both sides to eliminate square from the left side.

[00:03:59]So, here this square and this square root will be cancelled, and the value of AB, that is actually the diameter of this circle, that will become it will become square root of 52.

[00:04:16]And since here our target is to find out the radius of this circle, so therefore here we will divide this diameter by two.

[00:04:24]So, therefore our radius will become that will become diameter divided by two.

[00:04:33]And our diameter is simply square root of 52 divided by two.

[00:04:39]So, let's simplify this one number. Here we can write the square root of 52 as this 52 is same as 4 * 13, which is 52.

[00:04:49]Divided by two.

[00:04:52]And here we can write this expression in the numerator as square root of 4 * square root of 13. Here square root of 4 is simply 2 * square root of 13.

[00:05:03]Divided by two.

[00:05:05]Here we can cancel two with two.

[00:05:07]So, the final radius will become that will become square root of 13.

[00:05:15]So, that is our final answer.

[00:05:19]So, finally the radius of this circle, it is simply square root of 13 units.

[00:05:25]And that's our final target.