Practice (41)

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?

1142

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

$\dfrac{XQ}{QY}$ ?

1179

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is ,

$X$ in centimeters?

1191

A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?

1222

Angle

$ABC$ of

$\triangle ABC$ is a right angle. The sides of

$\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on

$\overline{AB}$ equals

$8\pi$ , and the arc of the semicircle on

$\overline{AC}$ has length

$8.5\pi$ . What is the radius of the semicircle on

$\overline{BC}$ ?

1224

A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are

$R_1 = 100$ inches,

$R_2 = 60$ inches, and

$R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?

1235

The ratio of the length to the width of a rectangle is

$4$ :

$3$ . If the rectangle has diagonal of length

$d$ , then the area may be expressed as

$kd^2$ for some constant

$k$ . What is

$k$ ?

1248

For some positive integers

$p$ , there is a quadrilateral

$ABCD$ with positive integer side lengths, perimeter

$p$ , right angles at

$B$ and

$C$ ,

$AB=2$ , and

$CD=AD$ . How many different values of

$p<2015$ are possible?

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?

1268

$\triangle{ABC}$ ,

$\angle{C} = 90^{\circ}$ 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?

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

$\triangle APQ$ ?

1301

In rectangle

$ABCD$ ,

$AB=20$ and

$BC=10$ . Let

$E$ be a point on

$\overline{CD}$ such that

$\angle CBE=15^\circ$ . What is

$AE$ ?

1309

Doug constructs a square window using

$8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is

$5 : 2$ , and the borders around and between the panes are

$2$ inches wide. In inches, what is the side length of the square window?

1325

Trapezoid

$ABCD$ has parallel sides

$\overline{AB}$ of length

$33$ and

$\overline {CD}$ of length

$21$ . The other two sides are of lengths

$10$ and

$14$ . The angles

$A$ and

$B$ are acute. What is the length of the shorter diagonal of

$ABCD$ ?

1332

Square

$ABCD$ has side length

$10$ . Point

$E$ is on

$\overline{BC}$ , and the area of

$\triangle ABE$ is

$40$ . What is

$BE$ ?

1341

$\triangle ABC$ ,

$AB=AC=28$ and

$BC=20$ . Points

$D,E,$ and

$F$ are on sides

$\overline{AB}$ ,

$\overline{BC}$ , and

$\overline{AC}$ , respectively, such that

$\overline{DE}$ and

$\overline{EF}$ are parallel to

$\overline{AC}$ and

$\overline{AB}$ , respectively. What is the perimeter of parallelogram

$ADEF$ ?

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

$BC$ ?

1354

All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

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

$AEDC$ ?

1377

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

$m+n$ ?

1381

A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?

1383

Let

$\angle ABC = 24^\circ$ and

$\angle ABD = 20^\circ$ . What is the smallest possible degree measure for angle CBD?

PlaneGeometry Triangle SimiarTriangle Circle AreaMethod CoordinatedGeometry GeometryPattern Combinatorics FactorizationMethod TrigInTriangle

AMC8 AMC10/12

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