Inscribed and Circumscribed Figures: Level 5 Challenges Practice Problems Online

Find the area of the brown colored region (outer region) if the radius of each circle is 4 units. Give your answer to 5 decimal places.

Suppose $ABCD$ is a cyclic quadrilateral with side $AD$ being a diameter of length $d$ , sides $AB$ and $BC$ both having length $a$ and side $CD$ having length $b$ such that $a,b,d$ are all positive integers with $a \ne b.$

Determine the minimum possible perimeter of $ABCD.$

An inscribed Hexagon has sides $AF = FE = ED = 4$ , $AB = BC = CD = 2$ . Furthermore, $ADEF$ and $ADCB$ are trapeziums.

If the length of chord $AD$ is $\frac{ m}{n}$ , where $m$ and $n$ are relatively prime integers, find $m + n$ .

In the figure there are three semicircles of diameters $AB$ , $AC$ , $BC$ and two circles $S_{1}$ and $S_{2}$ which are touching to the line $CF$ ( $CF$ is perpendicular to $AB$ ). Both circles have same radius and are called Archimedean Twins. Now if $AB = 10$ , $AC = 6$ and $BC = 4$ then find the radius of circle $S_{1}$ .

Let $ABCD$ be a cyclic quadrilateral with $\angle ABD = \angle ACD >90^{\circ}$ .

Let $P$ and $Q$ be the points on major arc $AD$ satisfying $PA=PC$ and $QB=QD$ .

Furthermore, let $X$ and $Y$ be the feet of the perpendiculars from $P$ and $Q$ to line $AD$ .

If $AB=20$ , $CD=14$ , and $BC=16$ , then find the length of $XY$ .

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Inscribed and Circumscribed Figures: Level 5 Challenges