Similar Triangles

A paper triangle with sides of lengths $3$, $4$, and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?

Rectangle $ABCD$ has $AB = 8$ and $BC = 6$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $\overline{AB}$ with $\overline{ME} \perp \overline{AC}$. What is the area of $\triangle AME$?

In rectangle $ABCD$, $AB = 5$ and $BC = 3$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF = 1$ and $GC = 2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

Right triangle $ABC$ has leg lengths $AB = 20$ and $BC = 21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$?

In $\triangle ABC$ with a right angle at $C$, point $D$ lies in the interior of $\overline{AB}$ and point $E$ lies in the interior of $\overline{BC}$ so that $AC = CD$, $DE = EB$, and the ratio $AC : DE = 4 : 3$. What is the ratio $AD : DB$?

In rectangle $ABCD$, $AB = 6$ and $BC = 3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE = EF = FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP : PQ : QD$ can be written as $r : s : t$ where the greatest common factor of $r$, $s$, and $t$ is $1$. What is $r + s + t$?

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$?

Rectangle $ABCD$ has $AB = 3$ and $BC = 4$. Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}$. What is the area of $\triangle AED$?

Triangle $ABC$ with $AB = 50$ and $AC = 10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC$. The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D$. Suppose $BP = 2$ and $PC = 3$. What is $AD$?

Trapezoid $ABCD$ has sides $AB = 92$, $BC = 50$, $CD = 19$, and $AD = 70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.

Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle ACB$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral with $AC = 2$. What is $BX^2$?

Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP = 11$, the length $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$?

In quadrilateral $ABCD$, $\angle BAD \cong \angle ADC$ and $\angle ABD \cong \angle BCD$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Triangle $ABC$ has $AB = 13$, $BC = 14$ and $AC = 15$. Let $P$ be the point on $\overline{AC}$ such that $PC = 10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE$?

Triangle $ABC$ with $AB = 50$ and $AC = 10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?

Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $\overline{AB}$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is $11/5$. Find the ratio of shaded region $D$ to the area of shaded region $A$.

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.

In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD} \perp \overline{BC}$, $\overline{DE} \perp \overline{AC}$, and $\overline{AF} \perp \overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

The altitude to the hypotenuse of a $30^\circ$-$60^\circ$-$90^\circ$ triangle is divided into two segments of lengths $x < y$ by the median to the shortest side of the triangle. What is the ratio $\frac{x}{x + y}$?

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 $\frac{x}{y}$?

In trapezoid $ABCD$, leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB = \sqrt{11}$ and $AD = \sqrt{1001}$, find $BC^2$.

Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB = 15$, $BQ = 20$, $PR = 30$, and $QS = 40$. Let $\frac{m}{n}$, in lowest terms, denote the perimeter of $ABCD$. Find $m + n$.

In the right triangle $\triangle ACE$, we have $AC = 12$, $CE = 16$, and $EA = 20$. Points $B$, $D$, and $F$ are located on $\overline{AC}$, $\overline{CE}$, and $\overline{EA}$, respectively, so that $AB = 3$, $CD = 4$, and $EF = 5$. What is the ratio of the area of $\triangle DBF$ to that of $\triangle ACE$?

A silo (a right circular cylinder) with diameter $20$ meters stands in a field. MacDonald is located $20$ meters west and $15$ meters south of the center of the silo. McGregor is located $20$ meters east and $g > 0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of $g$ can be written as $\frac{a\sqrt{b} - c}{d}$, where $a$, $b$, $c$, $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to $\gcd(a, c)$. What is $a + b + c + d$?

In rectangle $ABCD$, $AB = 1$, $BC = 2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?

Rectangle $ABCD$ has $AB = 4$ and $BC = 3$. Segment $\overline{EF}$ is constructed through $B$ so that $\overline{EF}$ is perpendicular to $\overline{DB}$, and $A$ and $C$ lie on $\overline{DE}$ and $\overline{DF}$, respectively. What is $EF$?

In rectangle $ABCD$, we have $AB = 8$, $BC = 9$, $H$ is on $\overline{BC}$ with $BH = 6$, $E$ is on $AD$ with $DE = 4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $\overline{GF} \perp \overline{AF}$. Find the length $GF$.

Square $ABCD$ has sides of length $1$. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD}$, respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. The length of a side of this smaller square is $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a + b + c$.