Practice (40)
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A semicircle is positioned above a square. The diameter of the semicircle is 2 units. We must find the radius, r, of the smallest circle that contains this figure.
Point $M$ of rectangle $ABCD$ is the midpoint of side $BC$ and point $N$ lies on $CD$ such that $DN:NC = 1:4$. Segment $BN$ intersects $AM$ and $AC$ at points $R$ and $S$. If $NS:SR:RB$ = $x:y:z$, what is the minimum possible value of $x + y + z$?
In $\triangle{ABC}$, segments AB and AC have each been divided into four congruent segments. We must find the fraction of the triangle that is shaded.
A semicircle and circle are placed inside a square with sides of length 4. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a side of the square (or 4). We must find the radius of the circle
The diameter of a spherical balloon is increased by 150%. We must find by what percent the volume increases.
The legs of a right triangle are in the ratio 3:4. One of the altitudes is 30 ft. What is the greatest possible area of this triangle?
Segments AD and BC are the radii of the top and bottom bases of the frustum. AD = 8, BC = 12 and AC = 15, what is the volume of the frustum?
We are asked to find the area of the shaded (grey) region where the distance between each dot is 1 cm.
A cylindrical container has a diameter of 8 cm (i.e., a radius of 4 cm) and a volume of 754 $cm^2$. Another container also has a diameter of 8 cm, but is twice as tall as the original container. We are asked to find the volume of the second container.
The angles of a triangle are in the ratio 1:3:5. What is the degree measure of the largest angle in the triangle?
In trapezoid $ABCD$, $AB = BC = 2AD$ and $AD= 5$. We are asked to find the area of trapezoid $ABCD$.
One line has a slope of \u22121/3 and contains the point (3, 6). Another line has a slope of 5/3 and contains the point (3, 0). We are asked to find the product of the coordinates of the point at which the two lines intersect.
Six circles of radius, $r = 1$ unit are drawn in the hexagon as shown. We must find the perimeter of the hexagon.
A circle is inscribed in a rhombus with sides of length 4cm. The two acute angles each measure $60^{\circ}$. We are asked to find the length of the circle'9s radius.
A line containing the points (-8, 9) and (-12, 12) intersects the $x$-axis at point $P$. Find the $x$-coordinate of point $P$.
Circle O has diameter AE and AE = 8. Point C is on the circumference of the circle such that segments AC and CE are congruent. Segment AC is a diameter of semicircle ABC and segment CE is a diameter semicircle CDE. What is the total combined area of the shaded regions?
The endpoints of a diameter of a circle are (-1, -4) and (-7, 6). We must find the coordinates of the center of the circle.
Line $l$ is perpendicular to the line with equation $6y$ = $kx +24$. The slope of line $l$ is $-2$. Find the value of $k$.
There is a shallow fish pond in the shape of a square. The perimeter of the pond is 24 ft and the water is 6 in deep. We must find the volume of the water in the pond.
The cube shown has a side length of $s$. Points $A$, $B$, $C$ and $D$ are vertices of the cube. We need to find the area of rectangle $ABCD$.
The diagram shows 8 congruent squares inside a circle. Find the ratio of the shaded area to the area of the circle.
A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$.
A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ .
In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $\angle AMN = \angle MNP = 90^{\circ}$. Compute the ratio $\frac{AP}{PB}$ .
A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons?