Similar Triangles: all problems

The diagram shows an octagon consisting of 10 unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base 5. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\dfrac{XQ}{QY}$?

Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC = CB = BA = 1$. The area of the "bat wings" (shaded area) is

Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD$, and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F$. The area of quadrilateral $AFED$ is $45$. What is the area of $ABCD$?

One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can fit into the remaining space?

A square piece of paper has side length $1$ and vertices $A, B, C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\triangle AEC'$?

Right $\triangle ABC$ has $AB = 3$, $BC = 4$, and $AC = 5$. Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $\overline{AC}$, $W$ on $\overline{AB}$, and $Z$ on $\overline{BC}$. What is the side length of the square?

Let $ABCDEFGHIJKL$ be the equilateral dodecagon shown below, and each angle is either $90^\circ$ or $270^\circ$. Let $M$ be the midpoint of $CD$, and suppose $HM$ splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

In trapezoid $ABCD$, points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $AB \parallel CD \parallel EF$. Given that $AB = 3$, $EF = 5$, and $CD = 6$, the ratio $[ABEF] / [CDEF]$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$. (Note: $[F]$ denotes the area of $F$.)

Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^\circ$. Given that $CP = 2$ and $CD = 3$, compute $CA$.

Let $ABC$ be a right triangle with hypotenuse $AB$ such that $AC = 36$ and $BC = 15$. A semicircle is inscribed in $ABC$ as shown, such that the diameter $XC$ of the semicircle lies on side $AC$ and that the semicircle is tangent to $AB$. What is the radius of the semicircle?

A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\dfrac{x}{y}$?