90% Failed to Solve This Math Question!

追加: 2026-05-07

[00:00:00]Given a semicircle inside a right triangle and AP is 8, CP is 9.

[00:00:16]And we have to find shaded area.

[00:00:22]So, in any circle from any point P, if we make two tangent, suppose this point is A and this point is B, then PA and PB, they will be equal.

[00:00:47]So, from point A, AP and AB, they will be equal.

[00:00:57]And AP is 8.

[00:01:00]That means AB, it will be also 8.

[00:01:07]And now, in triangle ABC, AB is 8, AC is 8 + 9, that is 17.

[00:01:23]So, we can find BC.

[00:01:32]BC squared, it will be AC squared minus AB squared.

[00:01:45]And BC squared is 17 squared minus 8 squared.

[00:01:56]And BC is squared is 289 minus 64.

[00:02:05]So, BC is squared is 225.

[00:02:12]That means BC it will be 15.

[00:02:20]BC is 15.

[00:02:28]Now, suppose the center of the semicircle is O and the radius is X.

[00:02:43]Then, BO it will be X and QO it will be X.

[00:02:53]And now, BC is 15. BO is X. QO is X. So, CQ it will be 15 minus 2X.

[00:03:11]And in any circle from any point P if PT is a tangent and AB is a chord, then PT is squared it is PA times PB.

[00:03:34]So, from point C CP is a tangent and QB is a chord.

[00:03:48]That means CP is it will be CQ times CB and CP is 9 is squared CQ is 15 minus 2x times CB is 15 and it is 81 is 15 minus 2x times 15 and 3 times 27 is 81 3 times 5 is 15 so 27 divided by 5 it is 15 minus 2x so 2x it will be 15 minus 27 divided by 5 and it is 75 minus 27 divided by 5 that will be 48 divided by 5 and 2x is 48 divided by 5 that means x will be 24 divided by 5 now shaded area it will be area of ABC minus area of semicircle and area of ABC it will be 1 over 2 times base is BC and height is AB minus area of semicircle it will be pi times radius is x s squared divided by 2 and it is 1 over 2 times BC is 15 AB is 8 minus pi over 2 times x is 24 divided by pi x is 24 divided by pi s squared and 2 times 4 is 8 so it is 60 minus pi over 2 times 576 divided by 25 and 2 times 288 is 576 so it will be 60 minus 288 pi divided by 25 and this is the final answer >> [music]