Use Blip to create an article or summary from any YouTube video.

I recently tackled a fascinating geometry problem that involved determining the radius of a circle. The problem was suggested by Mark, and it required a three-step solution.

The first step was to use right triangles. We began by constructing a radius from the center of the quarter circle to the endpoint of the 9-centimeter segment. Then, we constructed a rectangle and a right triangle with legs of 9, 16, and h, and a hypotenuse of R. Next, we constructed another radius from the center of the quarter circle to the endpoint of the 12-centimeter segment and a right triangle with legs of 9 + 12, h, and a hypotenuse of R. By equating the two equations, we found that H is equal to 13/4, and R is equal to 85/4 or 21.25.

The second method involved using coordinates. We set up a coordinate system with the center of the quarter circle at the point 0, 0. Then, we found the coordinates of two points on the circle and substituted them into the equation x^2 + y^2 = R^2. By solving the two equations, we found that R is equal to 85/4 or 21.25.

The third method used the chord chord power theorem or the intersecting chords theorem. We completed the entire circle and created chords of the larger circle. Then, we used the chord chord power theorem to solve for y and found that the length of the chord was equal to R + R, or 2R. By solving for R, we found that R is equal to 85/4 or 21.25.

The fourth method used geometry. We constructed an inscribed triangle in the larger circle and calculated the lengths of its three sides. Then, we used the formula for the circum radius to find that R is equal to 85/4 or 21.25.

In conclusion, the radius of the circle is 21.25 centimeters. This problem was a fun and challenging exercise in geometry, and I hope that it has sparked your interest in the subject.

The first step was to use right triangles. We began by constructing a radius from the center of the quarter circle to the endpoint of the 9-centimeter segment. Then, we constructed a rectangle and a right triangle with legs of 9, 16, and h, and a hypotenuse of R. Next, we constructed another radius from the center of the quarter circle to the endpoint of the 12-centimeter segment and a right triangle with legs of 9 + 12, h, and a hypotenuse of R. By equating the two equations, we found that H is equal to 13/4, and R is equal to 85/4 or 21.25.

The second method involved using coordinates. We set up a coordinate system with the center of the quarter circle at the point 0, 0. Then, we found the coordinates of two points on the circle and substituted them into the equation x^2 + y^2 = R^2. By solving the two equations, we found that R is equal to 85/4 or 21.25.

The third method used the chord chord power theorem or the intersecting chords theorem. We completed the entire circle and created chords of the larger circle. Then, we used the chord chord power theorem to solve for y and found that the length of the chord was equal to R + R, or 2R. By solving for R, we found that R is equal to 85/4 or 21.25.

The fourth method used geometry. We constructed an inscribed triangle in the larger circle and calculated the lengths of its three sides. Then, we used the formula for the circum radius to find that R is equal to 85/4 or 21.25.

In conclusion, the radius of the circle is 21.25 centimeters. This problem was a fun and challenging exercise in geometry, and I hope that it has sparked your interest in the subject.