Practice (40)

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3041
A circle inscribed in $\triangle{ABC}$ (the incircle) is tangent to $BC$ at $X$, to $AC$ at $Y$ , to $AB$ at $Z$. Show that $AX$, $BY$, and $CZ$ are concurrent.
3042
Three squares are drawn on the sides of $\triangle{ABC}$ (i.e. the square on $AB$ has $AB$ as one of its sides and lies outside $\triangle{ABC}$). Show that the lines drawn from the vertices $A, B, C$ to the centers of the opposite squares are concurrent.
3043
Let $P$ be a point inside a unit square $ABCD$. Find the minimal value of $AP+BP+CP$
3044
The diagonals $AC$ and $CD$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ such that $AM:AC = CN:CE=r$. Determine $r$ if $B, M,$ and $N$ are collinear.
3050

In rectangle $TUVW$, shown here, $WX = 4$ units, $XY = 2$ units, $YV = 1$ unit and $UV = 6$ units. What is the absolute difference between the areas of triangles $TXZ$ and $UYZ$.

3062
Rectangle $ABCD$ is shown with $AB = 6$ units and $AD = 5$ units. If $AC$ is extended to point $E$ such that $AC$ is congruent to $CE$, what is the length of $DE$?
3064

Diagonal $XZ$ of rectangle $WXYZ$ is divided into three segments each of length 2 units by points $M$ and $N$ as shown. Segments $MW$ and $NY$ are parallel and are both perpendicular to $XZ$. What is the area of $WXYZ$? Express your answer in simplest radical form.

3068

On line segment $AE$, shown here, $B$ is the midpoint of segment $AC$ and $D$ is the midpoint of segment $CE$. If $AD = 17$ units and $BE = 21$ units, what is the length of segment $AE$? Express your answer as a common fraction.

3074
A 12-sided game die has the shape of a hexagonal bipyramid, which consists of two pyramids, each with a regular hexagonal base of side length 1 cm and with height 1 cm, glued together along their hexagons. When this game die is rolled and lands on one of its triangular faces, how high off the ground is the opposite face? Express your answer as a common fraction in simplest radical form.
3075
Starting at the origin, a bug crawls 1 unit up, 2 units right, 3 units down and 4 units left. From this new point, the bug repeats this entire sequence of four moves 2015 more times, for a total of 2016 times. The coordinates of the bug’s final location are $(a, b)$. What is the value of $a + b$?
3076

A rectangular piece of cardboard measuring 6 inches by 8 inches is trimmed identically on all four corners, as shown, so that each trimmed corner is a quarter circle of greatest possible area. What is the perimeter of the resulting figure? Express your answer in terms of $\pi$.

3078
The areas of three faces of a rectangular prism are 54, 24 and 36 units . What is the length of the space diagonal of this prism? Express your answer in simplest radical form.
3079
Two 8-inch by 10-inch sheets of paper are placed flat on top of a 2-foot by 3-foot rectangular table. Nothing else is on the table, and the area of the table not covered by the sheets of paper is $708 in^2$. In square inches, what is the area of the overlap between the two sheets of paper?
3082

The polygon shown here is constructed from two squares and six equilateral triangles, each of side length 6 units. This polygon may be folded into a polyhedron by creasing along the dotted lines and joining adjacent edges as indicated by the arrows. What is the volume of the resulting polyhedron? Express your answer in simplest radical form.

3091

Quadrilateral $APBQ$, shown here, has vertices $A(0, 0)$ and $B(8, 0)$, and vertices $P$ and $Q$ lie on the line given by the equation $4x + 3y = 19$. If $PQ = 3$ units, what is the area of quadrilateral $APBQ$? Express your answer as a common fraction.

3093
Spring City is replanting the grass around a circular fountain in the center of the city. The fountain’s diameter is 10 feet, and the grass extends out from the edge of the fountain 20 feet in every direction. Grass seed is sold in bags that will each cover 300 $ft^2$ of grass. How many whole bags of grass seed will the city need to purchase?
3094

As shown, $ABCD$ is a square with side length equaling 10 cm, $CE\perp BE$, and $CE=8$. Find the area of the shaded triangle.
3095

As shown, a regular hexagon is inscribed in the bigger circle. If the area of the bigger circle is 2016 $cm^2$, find the total area of shaded regions.

3096

As shown, $D$ is the midpoint of $BC$. Point $E$ is on $AD$ such that $BE=AF$. Show that $AF=EF$.
3098
A 12-sided game die has the shape of a hexagonal bipyramid, which consists of two pyramids, each with a regular hexagonal base of side length 1 cm and with height 1 cm, glued together along their hexagons. When this game die is rolled and lands on one of its triangular faces, how high off the ground is the opposite face? Express your answer as a common fraction in simplest radical form.
3101

Rectangle $ABCD$ is shown with $AB = 6$ units and $AD = 5$ units. If $AC$ is extended to point $E$ such that $AC$ is congruent to $CE$, what is the length of $DE$?

3103

Let $ABCDE$ be a pentagon such that $AB=BC=CD=DE=EA$ as shown. If $\angle{ABC}=2\angle{DBE}$, find the measurement of $\angle{ABC}$.

3104

Let $ABCD$ be a trapezoid. Points $M$ and $N$ are the mid points of its diagonal $AC$ and $BD$, respectively. Show that $MN \parallel AB$ and $MN = \frac{1}{2}\mid AB - CD\mid$.

3105

A plane passing through the vertex $A$ and the center of its inscribed sphere of a tetrahedron $ABCD$ intersects its edge $BC$ and $CD$ at point $E$ and $F$, as shown. If $AEF$ divides this tetrahedron into two equal volume parts: $A-BDEF$ and $A-CEF$, what is the relationship between these two parts' surface areas $S_1$ and $S_2$ where $S_1 = S_{A-BDEF}$ and $S_1=S_{A-CEF}$? $(A) S_1 < S_2\quad(B) S_1 > S_2\quad (C) S_1 = S_2 \quad(D) $ cannot determine

3106
Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.
Sequence PlaneGeometry Triangle PythagoreanTheorem CevaTheorem MenalausTheorem TriangleCenter AreaMethod VectorMethod RotationMethod SolidGemoetry CoordinatedGeometry Trigonometry

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