Triangles: all problems

The twelve-sided figure shown has been drawn on $1\text{ cm}\times 1\text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$?

A triangle with vertices as $A = (1, 3)$, $B = (5, 1)$, and $C = (4, 4)$ is plotted on a $6\times 5$ grid. What fraction of the grid is covered by the triangle?

Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?

In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?

In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB = 12$, $BC = 4$, $CD = 3$, and $AD = 13$. What is the area of quadrilateral $ABCD$?

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?

In square $ABCE$, $AF = 2FE$ and $CD = 2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$?

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?

What is the area of the shaded pinwheel shown in the $5\times 5$ grid?

What is the area of the triangle formed by the lines $y = 5$, $y = 1 + x$, and $y = 1 - x$?

In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD : DC = 1 : 2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?

Right triangle $\triangle ABC$ has $AB = 5$, $BC = 12$, and $CA = 13$. Point $D$ lies on the angle bisector of $\angle BAC$ such that $CD$ is parallel to $AB$. Compute the length of $BD$.

In right triangle $\triangle ABC$ with $AB = 5$, $BC = 12$, and $CA = 13$, point $D$ lies on $CA$ such that $AD = BD$. The length of $CD$ can then be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.

What is the radius of a circle inscribed in a triangle with side lengths 6, 8, and 10?

A unit square contains four congruent non-overlapping equilateral triangles as shown in the figure. What is the largest possible side-length of one of the triangles?