Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 7"
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==Problem== | ==Problem== | ||
| − | If a diagonal <math>d</math> of a rectangle forms a <math>60^\circ</math> angle with one of its sides, then the area of the rectangle is | + | If a diagonal <math>d</math> of a [[rectangle]] forms a <math>60^\circ</math> [[angle]] with one of its sides, then the area of the rectangle is |
<math> \mathrm{(A) \ } \frac{d^2 \sqrt{3}}{4}\qquad \mathrm{(B) \ } \frac{d^2}{2}\qquad \mathrm{(C) \ } 2d^2\qquad \mathrm{(D) \ } d^2 \sqrt{2}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math> | <math> \mathrm{(A) \ } \frac{d^2 \sqrt{3}}{4}\qquad \mathrm{(B) \ } \frac{d^2}{2}\qquad \mathrm{(C) \ } 2d^2\qquad \mathrm{(D) \ } d^2 \sqrt{2}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math> | ||
==Solution== | ==Solution== | ||
| − | Using 30-60-90 triangle | + | Using <math>30-60-90</math> [[right triangle]] [[ratio]]s, the lengths of the sides of the rectangle are <math>\frac{d}{2}</math> and <math>\frac{d\sqrt{3}}{2}</math>. |
The area of the rectangle is <math>\frac{d^2\sqrt{3}}{4}\Rightarrow\mathrm{ A}</math>. | The area of the rectangle is <math>\frac{d^2\sqrt{3}}{4}\Rightarrow\mathrm{ A}</math>. | ||
==See also== | ==See also== | ||
| − | + | {{CYMO box|year=2007|l=Lyceum|num-b=6|num-a=8}} | |
| − | + | [[Category:Introductory Geometry Problems]] | |
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Revision as of 16:33, 6 May 2007
Problem
If a diagonal 
of a rectangle forms a 
angle with one of its sides, then the area of the rectangle is
Solution
Using 
right triangle ratios, the lengths of the sides of the rectangle are 
and 
.
The area of the rectangle is 
.
See also
| 2007 Cyprus MO, Lyceum (Problems) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
