All Questions ⁺⁰234662 Questions

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Tangents

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 15$ and $WX = 12,$ find $YZ.$

bader Jan 8, 2024

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18

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Circle

A circular table is pushed into a corner of the room, where two walls meet at a right angle. A point $P$ on the edge of the table (as shown below) has a distance of $1$ from one wall, and a distance of $1$ from the other wall. Find the radius read more ..

ElectricPavlov ●

bader Jan 8, 2024

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Tangents

Lines $XQ$ and $XR$ are tangent to a circle, as shown below. If $\angle QTA = 47^\circ$ and $\angle RUA = 65^\circ,$ then find $\angle QXR,$ in degrees.

bader Jan 8, 2024

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32

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Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{PA}$ is $4,$ and the circle has a radius of $4.$ Find the length $AB.$ read more ..

bader Jan 8, 2024

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Geometry

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $1$ and $BC = \sqrt{3}$, then find $AB$. read more ..

bader Jan 8, 2024

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10

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Circle

Points $T$ and $U$ lie on a circle centered at $O$, and point $P$ is outside the circle such that $\overline{PT}$ and $\overline{PU}$ are tangent to the circle. If $\angle TPO = 33^{\circ}$ and $PT = 10$, then what is the radius of the circle? read more ..

bader Jan 8, 2024

Jan 7, 2024

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Quarter-circle

A circle lies inside a quarter-circle, as shown below. The circle is tangent to side $\overline{AO}$ and arc $AB$ and side $\overline{BO}$. If the radius of the quarter-circle si 2, then find the radius of the circle. read more ..

bader Jan 7, 2024

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Chords of a circle

Chords $\overline{UV},$ $\overline{WX},$ and $\overline{YZ}$ of a circle are parallel. The distance between chords $\overline{UV}$ and $\overline{WX}$ is $1,$ and the distance between chords $\overline{WX}$ and $\overline{YZ}$ is also $1.$ If read more ..

bader Jan 7, 2024

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8

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Semicircle

A semicircle is inscribed in triangle $XYZ$ so that its diameter lies on $\overline{YZ}$, and is tangent to the other two sides. If $XY = 10,$ $XZ = 10,$ and $YZ = 10 \sqrt{2},$ then find the area of the semicircle. read more ..

bader Jan 7, 2024

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Quadrilateral

In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex quadrilateral $ABCD$ is $180.$ Find the area of the shaded region.
read more ..

tomtom Jan 7, 2024

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Tangent

Two circles are externally tangent at $T.$ The line $AB$ is a common external tangent to the two circles, and $P$ is the foot of the altitude from $P$ to line $AB.$ Find the length $TP.$

bader Jan 7, 2024

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Quarter-circle

A circle is inscribed in a quarter-circle, as shown below. If the radius of the quarter-circle is $4,$ then find the radius of the circle.

tomtom Jan 7, 2024

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24

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Polygon

Let $B,$ $A,$ and $D$ be three consecutive vertices of a regular $20$-gon. A regular heptagon is constructed on $\overline{AB},$ with a vertex $C$ next to $A.$ Find $\angle BAD,$ in degrees.

tomtom Jan 7, 2024

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Square

In the diagram, ABCD is a square. Find PR.

M, N, O, L are midpoints of sides.
AB = 12 read more ..

tomtom Jan 7, 2024

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Polygon

The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $4$ sides, then find the smallest angle, in degrees.

tomtom Jan 7, 2024

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Polygons

Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
An equilateral triangle, another equilateral triangle, two squares, and a regular $n$-gon, all with the same side read more ..

tomtom Jan 7, 2024

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Diagonals

The number of diagonals in a certain regular polygon is equal to $1/2$ times the number of sides. How many sides does this polygon have?

ElectricPavlov ●

tomtom Jan 7, 2024

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