Category Archives: Algebra

Algebra/Geometry Challenge – 2

Posted on December 8, 2024 by kevin

Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$, $AB=5$, $AD=8$. What is the length of the shorter diagonal of $ABCD$? Click here for the solution. Solution: Let $AC$ and $BD$ intersect at $E$, $x=BD$, $y=AE$, $z=CE$, $AC=y+z$. Because $BC=CD$, $AC$ is angle … Continue reading →

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Geometry Challenge – Area of Irregular Pentagon

Posted on October 15, 2024 by kevin

As shown in the diagram below, a star sign consists of five straight lines. It produces five triangles and a pentagon. If areas of five triangles are 3, 10, 7, 15, and 8 square unit respectively. Find the area of … Continue reading →

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Geometry Probability – 3

Posted on September 18, 2024 by kevin

Two points are randomly and uniformly selected from the interior of a circle. The center of the circle and the two points joined together form a triangle. What is the probability that the triangle is acute? Click here for the … Continue reading →

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Geometry Probability – 2

Posted on August 5, 2024 by kevin

An iron rod of $1$ foot long is cut into three segments with random length. (1) What is the probability that the three segments form a triangle? (2) What is the probability that the three segments form an acute triangle? … Continue reading →

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Geometry Probability – 1

Posted on July 22, 2024 by kevin

3 random points are chosen independently on a circle. What is probability that they form an acute triangle? Click here for the solution. Solution: This problem is equivalent to the following: Two cuts are randomly made on a line of … Continue reading →

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