Circles with centers $O$ and $P$ have radii $2$ and $4$, respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points $C$ and $D$ on the circle with center $P$ are such that $AD$ and $BC$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$?
A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?
In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50}$ and $BE$=1. What is the area of the inner square $EFGH$?
The figure shown is called a trefoil and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $2$?
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed?
An equiangular octagon has four sides of length 1 and four sides of length $\frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC \perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?
In $ABC$ we have $AB = 25$, $BC = 39$, and $AC=42$. Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$. What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$?
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?
An $8$-foot by $10$-foot floor is tiles with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $\frac{1}{2}$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$?
Equilateral $\triangle ABC$ has side length $2$, $M$ is the midpoint of $\overline{AC}$, and $C$ is the midpoint of $\overline{BD}$. What is the area of $\triangle CDM$?
In trapezoid $ABCD$ we have $\overline{AB}$ parallel to $\overline{DC}$, $E$ as the midpoint of $\overline{BC}$, and $F$ as the midpoint of $\overline{DA}$. The area of $ABEF$ is twice the area of $FECD$. What is $\frac{AB}{DC}$?
In square ABCD, shown here, sector BCD was drawn with a center C and BC = 24 cm. A semicircle with diameter AE is drawn tangent to the sector BCD. If points A, E and D are collinear, what is AE?
What is the greatest possible area of a triangle with vertices on or above the $x$-axis and on or below the parabola $y = -(x\u200a - \frac{1}{2})^2+ 3$? Express your answer in simplest radical form.
This figure consists of eight squares labeled A through H. The area of square F is16 units$^2$. The area of square B is 25 units$^2$. The area of square H is 25 units$^2$. In square units, what is the area of square D?
Points D, E and F lie along the perimeter of $\triangle ABC$ such that $\overline{AD}$ , $\overline{BE}$ and $\overline{CF}$ intersect at point G. If AF = 3, BF = BD = CD = 2 and AE = 5, then what is $\frac{BG}{EG}$ ? Express your answer as a common fraction.
An optometrist has this logo on his storefront. The center circle has area 36\u03c0 $in^2$, and it is tangent to each crescent at its widest point (A and B). The shortest distance from A to the outer circle is $\frac{1}{3}$ the diameter of the smaller circle. What is the area of the larger circle? Express your answer in terms of \u03c0.
Quadrilateral ABCD is a square with BC = 12 cm. $\overset{\frown} {BOC}$ and $\overset{\frown} {DOC}$ are semicircles. what is the area of the shaded region?
If the point $(x, x)$ is equidistant from (-2, 5) and (3, -2), what is the value of $x$?
The two cones shown have parallel bases and common apex $T$. $TW = 32$ m, $WV = 8$ m and $ZY = 5$ m. What is the volume of the frustum with circle $W$ and circle $Z$ as its bases? Express your answer in terms of $\pi$.
If $f(x) = 3x^2$, what is the x-coordinate of the point of intersection of the graphs of $y = f(x)$ and $y = f(x \u2212 4)$?
In isosceles trapezoid $ABCD$, shown here, $AB = 4$ units and $CD = 10$ units. Points $E$ and $F$ are on $\overline{CD}$ with $\overline{BE}$ parallel to $\overline{AD}$ and $\overline{AF}$ parallel to $\overline{BC}$. $\overline{AF}$ and $\overline{BE}$ intersect at point $G$. What is the ratio of the area of triangle $EFG$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.
In square units, what is the largest possible area a rectangle inscribed in the triangle shown here can have?
A line segment with endpoints A(3, 1) and B(2, 4) is rotated about a point in the plane so that its endpoints are moved to A' (4, 2) and B' (7, 3), respectively. What are the coordinates of the center of rotation? Express your answer as an ordered pair.