All Questions ⁺⁰230874 Questions

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Geometry

Find the area of triangle $ABC$ if $AB = 6,$ $BC = 8,$ and $\angle ACB = 30^\circ$.

Pythagorearn Apr 22, 2024

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geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $AX = 5,$ and $CX = 5$. What is $BX$?

Pythagorearn Apr 22, 2024

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Geometry

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 14$, $AY = 14$, and $CY = 8$, find $BZ$.

Pythagorearn Apr 22, 2024

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Geometry

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 14$, $AY = 14$, and $CY = 8$, find $BC$.

Pythagorearn Apr 22, 2024

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Geometry

Find the circumradius of triangle ABC.

Pythagorearn Apr 22, 2024

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Geometry

Find the inradius of triangle ABC.

Pythagorearn Apr 22, 2024

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Geometry

Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side length of the original cube is $6$, then find the volume of the resulting read more ..

Pythagorearn Apr 22, 2024

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Geometry

In triangle $ABC$, $AB = 7$, $AC = 17$, and the length of median $AM$ is $12$. Find the area of triangle $ABC$.

Pythagorearn Apr 22, 2024

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Geometry

In triangle $ABC$, $AB = 10$ and $AC = 15$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 1:3$, then find $AD$.

Pythagorearn Apr 22, 2024

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Geometry

We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude to $\triangle ABC$. If $AC = 8,$ $BK = 2$, and $CK = 3,$ then what is $AB$?

Pythagorearn Apr 22, 2024

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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 $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ read more ..

Pythagorearn Apr 22, 2024

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Geometry

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

Pythagorearn Apr 22, 2024

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Geometry

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 of6 from one wall, and a distance of 6 from the other wall. Find the radius of read more ..

Pythagorearn Apr 22, 2024

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Geometry

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 $XY = 10$ and $WX = 8,$ find $YZ.$

Pythagorearn Apr 22, 2024

Apr 21, 2024

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Geometry (Inradius and Incircles)

1) I is the incenter of triangle ABC. If angle ABC = 78^\circ$ and angle ACB = 47^\circ,$ find angle BID, in degrees.

2) Find angle read more ..

Pythagoream Apr 21, 2024

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Geometry

In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.

Akhain1 Apr 21, 2024

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Geometry

Quadrilateral ABCD is a parallelogram. Let E be a point on \overline{AB}, and let F be the intersection of lines DE and BC. The area of triangle EBF is 1, and the area of triangle BCE is 2. Find the area of parallelogram ABCD.

Akhain1 Apr 21, 2024

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Geometry

In trapezoid ABCD, \overline{AB} \parallel \overline{CD}. Find the area of the trapezoid.

Akhain1 Apr 21, 2024

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Geometry

In trapezoid PQRS, \overline{PQ} \parallel \overline{RS}. Let X be the intersection of diagonals \overline{PR} and \overline{QS}. The area of triangle PRS is 27, and the area of triangle PQR is 75. Find the area of trapezoid PQRS read more ..

Akhain1 Apr 21, 2024

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Geometry

In parallelogram EFGH, let M be the point on \overline{EF} such that FM:ME = 2:1, and let N be the point on \overline{EH} such that HN:NE = 3:8. Line segments \overline{FH} and \overline{GM} intersect at P, and line segments \overline{FH} and \overline{GN} read more ..

Akhain1 Apr 21, 2024

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Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $AMC$ if $[ABC] = 14$. read more ..

Akhain1 Apr 21, 2024

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Geometry

Points $M$, $N$, and $O$ are the midpoints of sides $\overline{KL}$, $\overline{LJ}$, and $\overline{JK}$, respectively, of triangle $JKL$. Points $P$, $Q$, and $R$ are the midpoints of $\overline{NO}$, $\overline{OM}$, and $\overline{MN}$, respectively. read more ..

Akhain1 Apr 21, 2024

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Geometry

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

Akhain1 Apr 21, 2024

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help geometry

Altitudes AD and BE of acute triangle ABC intersect at point H. If angle ABH = 30 and angle DAC = 45, then what is angle HCA in degrees? read more ..

hasAquestion ●

Akhain1 Apr 21, 2024

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Counting

A speech class has $4$ freshmen and $3$ sophomores. (Everyone is distinguishable.) In how many ways can they stand in line, so that all the freshmen are standing together?

magenta Apr 21, 2024

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