The radius inscribed in a right-angled triangle of a circle is equal to half
| July 25, 2021 | education
The radius inscribed in a right-angled triangle of a circle is equal to half the difference between the legs. Find the ratio of the larger leg to the smaller leg.
By condition, R = (AC – AB) / 2.
Since the circle is inscribed in a right-angled triangle, then R = (AC + AB – BC) / 2.
By the Pythagorean theorem, BC = √ (AC ^ 2 + AB ^ 2).
Then (AC – AB) / 2 = (AC + AB – √ (AC ^ 2 + AB ^ 2)) / 2.
2 * AB = √ (AC ^ 2 + AB ^ 2).
4 * AB ^ 2 = AC ^ 2 + AB ^ 2.
3 * AB ^ 2 = AC ^ 2.
AC ^ 2 / AB ^ 2 = 3.
AC / AB = √3.
Answer: The leg ratio is √3 / 1.
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