Practice (42)

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

$\triangle ABC$ .

371

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

$b$ ?

397

$\triangle ABC$ ,

$\angle C = 90^\circ$ 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?

407

What is the greatest possible perimeter of an isosceles triangle with sides of length

$5x + 20$ ,

$3x + 76$ and

$x + 196$ ?

413

What is the radius of a circle inscribed in a triangle with sides of length $5$ , $12$ and $13$ units?

488

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

$a+b+c$ ?

606

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

$\triangle AMN?$

731

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

$BC$ ?

950

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?

1062

Two angles of an isosceles triangle measure

$70^\circ$ and

$x^\circ$ . What is the sum of the three possible values of

$x$ ?

1138

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?

1262

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?

1271

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$ . What is $FG + JH + CD$ ?

1288

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?

1293

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

$\triangle APQ$ ?

1344

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?

1352

$\triangle ABC$ ,

$AB = 86$ , 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

$BC$ ?

1370

In triangle

$ABC$ , medians

$AD$ and

$CE$ intersect at

$P$ ,

$PE=1.5$ ,

$PD=2$ , and

$DE=2.5$ . What is the area of

$AEDC$ ?

1468

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

$BD$ ?

1500

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?

1544

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

$\triangle ABC$ ?

1569

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

$x$ ?

1602

A right triangle has perimeter

$32$ and area

$20$ . What is the length of its hypotenuse?

1648

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?

1676

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

$AB$ ?

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