Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$?
In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
What is the greatest possible perimeter of an isosceles triangle with sides of length $5x + 20$, $3x + 76$ and $x + 196$?
What is the radius of a circle inscribed in a triangle with sides of length $5$, $12$ and $13$ units?
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?
Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$
Triangle $ABC$ has $AB = 13$ and $AC = 15$, and the altitude to $\overline{BC}$ has length $12$. What is the sum of the two possible values of $BC$?
Bill walks $\tfrac12$ mile south, then $\tfrac34$ mile east, and finally $\tfrac12$ mile south. How many miles is he, in a direct line, from his starting point?
Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$?
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is $30\%$ of the perimeter. What is the length of the longest side?
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG + JH + CD$?
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)
The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle?
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?
Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?
In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$?
In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC$?
The area of $\triangle$$EBD$ is one third of the area of $3-4-5$ $\triangle$$ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$?
Nondegenerate $\triangle ABC$ has integer side lengths, $\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. What is the smallest possible value of the perimeter?
Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=3$, and $DC=4$. What is the area of $\triangle ABC$?
A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$?
A right triangle has perimeter $32$ and area $20$. What is the length of its hypotenuse?
A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?
Point $P$ is inside equilateral $\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$?