In a right triangle ABC with AB = x, BC = 50, and a circle with center O and radius 5 tangent to AB at P and BC at Q, the length x can be found using the properties that radii are perpendicular to tangents (forming a square POQB with side 5), tangent segments from a point are equal (AP = AS = x-5, CQ = CS = 45), and the Pythagorean theorem (x² + 50² = (x+40)²), which simplifies to x = 45/4.
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Can you find the length X in the right triangle?Added:
Can you find the length x?
Suppose the center of the circle is O.
And suppose this point is P.
This point is Q.
And this point is S.
And we have AB is x.
BC is 50.
And now if we join OP and OQ then OP is radius. That will be five.
And OQ is also radius.
That will be five.
And in any circle radius and tangent they are perpendicular on each other.
That means OP it will be perpendicular on AB.
And OQ it will be perpendicular on BC.
So POQB it will be a square.
And PO is five.
OQ is five.
That means BQ it will be also five.
And BP it will be also five.
And BC is 50.
BQ is five.
That means CQ will be 45.
And AB is X.
BP is five.
So AP it will be X minus five.
And Now in any circle from any point P if we make two tangent on the circle suppose this point is A and this point is B then PA and PB they will be equal.
So from point A AP is X minus five.
That means AS it will be also X minus five.
And from point C CQ is 45.
So, C is It will be also 45.
And now AB is squared plus BC is squared It will be AC is squared.
And AB is X is squared.
BC is 50 is squared.
And AC is X minus 5 plus 45.
is squared And X is squared plus 50 is squared is X plus 40 is squared.
And It is X is squared plus 2500 is X is squared plus 40 is squared is 1600 plus two times X times 40 is 80X.
And X is squared will get cancelled.
So, 2500 is 1600 plus 80X.
And 2500 minus 1600 is 80X.
And 900 is 80x.
So, x will be 900 divided by 80.
And zero will get cancelled.
And two times 45 is 90.
Two times four is eight.
So, it is 45 divided by four.
So, x is 45 divided by four.
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