The lines $x=\frac{1}{4}y+a$ and $y=\frac{1}{4}x+b$ intersect at the point $(1,2)$. What is $a+b$?
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 $AOBCPD$?
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?
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 $\triangle ABC$, 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 $BD$?
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 $\frac{AB}{DC}$?
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.
In right $\triangle{ABC}$, shown here, AC = 24 units and BC = 7 units. Point D lies on $\overline{AB}$ so that $\overline{CD} \perp \overline{AB}$. The bisector of the smallest angle of $\triangle{ABC}$ intersects $\overline{CD}$ at point E. What is the length of $\overline{ED}$ ? Express your answer as a common fraction.
Two equilateral triangles are drawn in a square, as shown. In degrees, what is the measure of each obtuse angle in the rhombus formed by the intersection of the two triangles?
The perimeter of a rectangle is 22 cm and its area is 24 $cm^2$. What is the smaller of the two integer dimensions of the rectangle?
The analog clock shown has a minute hand with an arrow tip that is exactly twice as far from the clock\u2019s center as the hour hand\u2019s arrow tip. If point A is at the tip of the minute hand, and point B is at the tip of the hour hand, what is the ratio of the distance that point B travels in 3 hours to the distance that point A travels in 9 hours? Express your answer as a common fraction.
A square and a regular hexagon are coplanar and share a common side as shown. What is the sum of the degree measures of angles 1 and 2?
Triangle $\triangle{MNO}$ is an isosceles trianglewith MN = NO = 25. A line segment drawn from the midpoint of MO perpendicular to MN, intersects MN at point P with NP:PM = 4:1. We must find the length of the altitude drawn from point N to side MO.
In rectangle $ABCD$, $AB = 6$ units. m$\angle{DBC} = 30^{\circ}$, $M$ is the midpoint of segment $AD$, and segments $CM$ and $BD$ intersect at point $K$. We must find the length of segment $MK$.
A right triangle has sides with lengths 8, 15 and 17. A circle is inscribed in the triangle, as shown, and we must find the radius of the circle.
A triangle has angles measuring $15^{\circ}$, $45^{\circ}$ and $120^{\circ}$. The side opposite the $45^{\circ}$ angle is 20 units. The area of the triangle can be expressed as $m -n\sqrt{q}$ and we must find the sum $m + n + q$.
A semicircle is positioned above a square. The diameter of the semicircle is 2 units. We must find the radius, r, of the smallest circle that contains this figure.
Point $M$ of rectangle $ABCD$ is the midpoint of side $BC$ and point $N$ lies on $CD$ such that $DN:NC = 1:4$. Segment $BN$ intersects $AM$ and $AC$ at points $R$ and $S$. If $NS:SR:RB$ = $x:y:z$, what is the minimum possible value of $x + y + z$?
A semicircle and circle are placed inside a square with sides of length 4. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a side of the square (or 4). We must find the radius of the circle
The legs of a right triangle are in the ratio 3:4. One of the altitudes is 30 ft. What is the greatest possible area of this triangle?
The angles of a triangle are in the ratio 1:3:5. What is the degree measure of the largest angle in the triangle?
Six circles of radius, $r = 1$ unit are drawn in the hexagon as shown. We must find the perimeter of the hexagon.
A circle is inscribed in a rhombus with sides of length 4cm. The two acute angles each measure $60^{\circ}$. We are asked to find the length of the circle'9s radius.
A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ .
In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $\angle AMN = \angle MNP = 90^{\circ}$. Compute the ratio $\frac{AP}{PB}$ .