Practice (40)

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1835

Let $\triangle ABC$ be a right triangle whose three sides' lengths are all integers. Prove among its three sides' lengths, at lease one is a multiple of $3$, one is a multiple of $4$, and one is a multiple of $5$. (Note: they can be the same side. For example, in the $5-12-13$, $12$ is both a multiple of $3$ and $4$.)
1839

In rectangle ABCD, BC = 2AB. Points O and M are the midpoints of $\overline{AD}$ and $\overline{BC}$ , respectively. Point P bisects $\overline{AO}$ . If OB = $6\sqrt{2}$ units, what is the area of $\triangle{NOP}$?

1841
What is the length of the shortest segment that can be drawn from the point (4, 1) to 2x - y + 4 = 0? Express your answer as a decimal to the nearest hundredth.
1843

In right $\triangle{ABC}$, shown here, AC = 24 units and BC = 7 units. Point D lies on $\overline{AB}$ so that $\overline{CD} \perp \overline{AB}$. The bisector of the smallest angle of $\triangle{ABC}$ intersects $\overline{CD}$ at point E. What is the length of $\overline{ED}$ ? Express your answer as a common fraction.

1847

A square prism has dimensions $5' \times 5' \times 10'$, where ABCD is a square. AP = ER = 2 ft and QC = SG = 1 ft. The plane containing $\overline{PQ}$ and $\overline{RS}$ slices the original prism into two new prisms. What is the volume of the larger of these two prisms?

1853

A square of side length 1 inch is drawn with its center A on a circle O of radius 1 inch such that a side of the square is perpendicular to $\overline{OA}$ , as shown. What is the area of the shaded region? Express your answer as a decimal to the nearest hundredth.

1865

Two equilateral triangles are drawn in a square, as shown. In degrees, what is the measure of each obtuse angle in the rhombus formed by the intersection of the two triangles?

1867
The perimeter of a rectangle is 22 cm and its area is 24 $cm^2$. What is the smaller of the two integer dimensions of the rectangle?
1872
A line passes through the points (-2, 8) and (5, -13). When the equation of the line is written in the form $y = mx + b$, what is the product of $m$ and $b$?
1873

Mr. Mayfeld is designing a sign for his ice cream shop. The sign will be a shape consisting of a semicircle and an isosceles triangle that he will paint to look like a cone with a scoop of ice cream. He will cut the figure out of a rectangular piece of plywood measuring 2 ft by 4 ft, as shown. The shaded regions will be cut away. If BE = 3BG and $\overline{AB}$ is parallel to $\overline{CE}$ , what is the total area of the resulting figure? Express your answer as a decimal to the nearest tenth.

1875

A right square pyramid has a base with a perimeter of 36 cm and a height of 12 cm. At one-third of the distance up from the base to the apex, the pyramid is cut by a plane parallel to its base. What is the volume of the top pyramid?

1878
A right rectangular prism has a volume of 720 $cm^3$. Its surface area is 484 $cm^2$. If all edge lengths are integers, what is the length of the longest segment that can be drawn that connects two vertices? Express your answer in simplest radical form.
1883

The analog clock shown has a minute hand with an arrow tip that is exactly twice as far from the clock\u2019s center as the hour hand\u2019s arrow tip. If point A is at the tip of the minute hand, and point B is at the tip of the hour hand, what is the ratio of the distance that point B travels in 3 hours to the distance that point A travels in 9 hours? Express your answer as a common fraction.

1885

In the figure shown here, the distance between any two horizontally or vertically adjacent dots is one unit. What is the area of the shaded polygon? Express your answer as a decimal to the nearest tenth.

1892

A square is inscribed in a circle of radius 5 units. In each of the four regions bounded by a side of the square and the smaller circular arc joining the endpoints of that side, a square is drawn so that one side lies on the side of the larger square and the two opposite vertices lie on the circle, as shown. What is the total area of the five squares? Express your answer to the nearest whole number.

1894

A square and a regular hexagon are coplanar and share a common side as shown. What is the sum of the degree measures of angles 1 and 2?

1896
$\triangle{ABC}$ has vertices at A(-3, 4), B(5, 0) and C(1, -4). What is the $x$-coordinate of the point where the median from C intersects $\overline{AB}$?
1907
A pyramid has 6 vertices and 6 faces. How many edges does it have?
1910
Let's name the coordinates of the vertices of a trapezoid are A(1, 7), B(1, 11), C(8, 4) and D(4, 4). What is the area of the trapezoid?
1913
Triangle $\triangle{MNO}$ is an isosceles trianglewith MN = NO = 25. A line segment drawn from the midpoint of MO perpendicular to MN, intersects MN at point P with NP:PM = 4:1. We must find the length of the altitude drawn from point N to side MO.

1918
In rectangle $ABCD$, $AB = 6$ units. m$\angle{DBC} = 30^{\circ}$, $M$ is the midpoint of segment $AD$, and segments $CM$ and $BD$ intersect at point $K$. We must find the length of segment $MK$.
1921
A rectangular prism is composed of unit cubes. The outside faces of the prism are painted blue and the seven unit cubes in the interior are unpainted. We must find how many unit cubes have exactly one painted face.
1923
A right triangle has sides with lengths 8, 15 and 17. A circle is inscribed in the triangle, as shown, and we must find the radius of the circle.

1925
In trapezoid ABCD segments AB and CD are parallel. Point P is the intersection of diagonals AC and BD. The area of $\triangle{PAB}$ is 16 and $\triangle{PCD}$ is 25. We must find the area of the trapezoid.
1927
A triangle has angles measuring $15^{\circ}$, $45^{\circ}$ and $120^{\circ}$. The side opposite the $45^{\circ}$ angle is 20 units. The area of the triangle can be expressed as $m -n\sqrt{q}$ and we must find the sum $m + n + q$.
PlaneGeometry Triangle AreaMethod SolidGemoetry CoordinatedGeometry PythagoreanTripletFormula



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