falafronk

94 Questions
0 Answers

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28

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Altitudes

Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 12,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$

falafronk Mar 27, 2024

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26

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+359

Quadratic

Fill in the blanks to make a quadratic whose roots are 5 and -5.
x^2 + ___ x + ___

falafronk Mar 26, 2024

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30

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+359

Solid 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 ..

falafronk Mar 26, 2024

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35

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+359

Pyramids

Let $ABCD$ be a regular tetrahedron. Let $E$, $F$, $G,$ $H$ be the centers of faces $BCD$, $ACD$, $ABD$, $ABC$, respectively. The volume of pyramid $DEFG$ is $18.$ Find the volume of pyramid $EFGH$

falafronk Mar 26, 2024

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42

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+359

help with geometry

Let $APQRS$ be a pyramid, where the base $PQRS$ is a square. The total surface area of pyramid $APQRS$ (including the base) is $800$. Let $W$, $X$, $Y$, and $Z$ be the midpoints of $\overline{AP}$, $\overline{AQ}$, $\overline{AR}$, and $\overline{AS}$, read more ..

falafronk Mar 26, 2024

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19

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Circle

A circle lies inside a quarter-circle, as shown below. The circle is tangent to side AO and arc AB. Find the radius of the circle.

falafronk Mar 26, 2024

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23

1

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help hard geometry

In triangle ABC, point X is on side BC such that AX=5, BX=5, CX=4, and the circumcircles of triangles ABX and ACX have the same radius. Find the area of triangle ABC.

falafronk Mar 26, 2024

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38

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+359

Semicircles

Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.

falafronk Mar 26, 2024

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30

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+359

Pyramid

Let $ABCDE$ be a right square pyramid, with base $ABCD$ and apex $E.$ Find the volume of the pyramid.

falafronk Mar 25, 2024

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21

1

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Prism

Let $ABCDEFGH$ be right rectangular prism. The total surface area of the prism $15.$ Also, the sum of all the edges of the prism is $17.$ Find the length of the diagonal joining one corner of the prism to the opposite corner.

falafronk Mar 25, 2024