tag:blogger.com,1999:blog-38193052.post8009494475879610202..comments2024-03-20T08:57:17.447-03:00Comments on Jornalheiros: Geometria – Três quadrados em um círculoPC Filhohttp://www.blogger.com/profile/16547063456626761789noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-38193052.post-55491348274032406802020-03-23T02:34:21.572-03:002020-03-23T02:34:21.572-03:00Perfect, Jake. A beautiful analytic geometry class...Perfect, Jake. A beautiful analytic geometry class. :)PC Filhohttps://www.blogger.com/profile/16547063456626761789noreply@blogger.comtag:blogger.com,1999:blog-38193052.post-64984451644000990782020-03-22T16:59:22.773-03:002020-03-22T16:59:22.773-03:00Time to break out our old friend analytic geometry...Time to break out our old friend analytic geometry!<br /><br />We put the circle on a coordinate plane so that the midpoint of the top side of the largest square is at the origin. This means that the top vertices of the top square are (±3,18) and the bottom vertices of the bottom square are (±9,-18). From the symmetry of these four vertices we conclude that the x-coordinate of the center of the circle is 0. That means an equation of the circle is x^2 + (y-k)^2 = r^2, where k is the still-unknown y-coordinate of the center and r is the radius. We now solve for k and r by plugging in the coordinates of one top vertex and one bottom vertex into this equation:<br /><br />3^2 + (18-k)^2 = r^2 -> (k-18)^2 = r^2 - 9 (Equation 1)<br />9^2 + (-18-k)^2 = r^2 -> (k+18)^2 = r^2 - 81 (Equation 2)<br /><br />Subtracting equation 2 from equation 1, we get:<br /><br />2k*-36 = 72<br /><br />or<br /><br />k = -1<br /><br />Therefore the center of the circle is at (0,-1), or 1 unit below the midpoint of the top side of the largest square.<br /><br />We now plug k = -1 back into either equation 1 or equation 2:<br /><br />(-1+18)^2 = r^2 - 81<br /><br />or<br /><br />r^2 = 370<br /><br />This gives us r^2. Since the area of a circle is π times r^2, that means the area of the circle is 370π square units.<br />jrh150482https://www.blogger.com/profile/10502831081969372299noreply@blogger.com