Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

1997 PMWC Problems/Problem T1

Revision as of 10:23, 24 August 2008 by 1=2 (talk | contribs) (New page: ==Problem== Let <math>PQR</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</m...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Let $PQR$ be an equilateral triangle with sides of length three units. $U$, $V$, $W$, $X$, $Y$, and $Z$ divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral $UWXY$ to the area of the triangle $PQR$.

1997 PMWC team 1.png

Solution

Triangles UWQ, PUY, UWX, and UXY are all right triangles with side lengths 1, $\sqrt{3}$, and 2. Thus $[UWXY]=\sqrt{3}$ and $[PQR]=\frac{9}{4}\sqrt{3}$. $\frac{[UWXY]}{[PQR]}=\frac{1}{\frac{9}{4}}=\boxed{\frac{4}{9}}$

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems/Problem_T1&oldid=27559"