Practice (64)

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A line containing the points (-8, 9) and (-12, 12) intersects the $x$-axis at point $P$. Find the $x$-coordinate of point $P$.

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 $k$.

The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at (0, 4) and fires a cannonball in a straight line at the closest point on the wall. Compute the $y$-coordinate of the point on the wall that the cannonball hits.

The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 -37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair.

What is the area of region bounded by the graphs of $y=|x+2| -|x-2|$ and $y=|x+1|-|x-3|$?

In the diagram $ABCDEFG$ is a regular heptagon (a 7 sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. %


Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$?

Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$?

Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that $$\frac{x_1 + x_2 + x_3 + x_44}{4}$$ is independent of the line, and compute its value.

In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$?

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?

All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?

A dilatation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation?

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$

A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?

In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?

Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$.

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.

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.

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.

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 $DE$?

Let $AB=2$ is a diameter of circle $O$. If $AC=AO$, $AC\perp AB$, $BD=\frac{3}{2}\cdot AB$, $BD\perp AB$ and $P$ is a point on arc $AB$. Find the largest possible area of the enclosed polygon $ABDPC$.

Two circles, $O_1$ and $O_2$ are tangent. Let $AB$ be their common tangent line which touches $O_1$ at point $A$ and touches $O_2$ at point $B$. Extend $AO_1$ and intersects $O_1$ at another point $C$. Line $CD$ is tangent to circle $O_2$ at point $D$. Show that $AC=CD$.