Remember when you were a kid and the box of Lucky Charms had a toy buried inside? Yeah, hold on to that memory, because thanks to advertising laws, most cereal manufacturers stopped offering such promotions in 2008. I sure wish law makers would at least allow cereal manufacturers to put toys in cereals with low sugar content, as I can't get my twins to even listen to the Snap! Crackle! and Pop! of Rice Krispies. But I guess that's for another blog on another day.
A box of cereal with a toy inside was so much more appealing than the lame cereal on the same shelf with no promise of a surprise inside of every box. It didn't matter that the toy might be a five cent ring that broke easily or a crummy plastic top that didn't exactly spin--it was a toy! And a surprise! And if the stars aligned just right over my breakfast table, that cheap packet of joy fell into my bowl instead of my sisters' bowls.
Equilateral triangles are sort of like that cereal box. There is a surprise in every one, and if you find it, you'll often earn points on the ACT and SAT that other test takers will miss.
Every equilateral triangle contains two hidden 30º:60º:90º triangles:
When provided with just a single side length of an equlilateral triangle, we can find its height using the 30º:60º:90º relationship (this relationship is provided in the formula box on the SAT but must be memorized for the ACT). Once you have the height of the triangle, it's easy to determine the area.
Let's see how this secret can help us on the ACT and the SAT:
The height of an equilateral triangle is also the leg of a hidden 30º:60º:90º triangle. DIAGRAM the question:
As you can see, the height of the triangle is 4 square root 3. Now, use the formula for the area of a triangle:
Answer choice (C) is correct. This little secret takes an otherwise difficult question and gives you an easy way to solve it in under ten seconds. It's like geometry's version of the cereal box, but more valuable and less regulated by the government.
Have more questions about the secrets hidden in the ACT? Check out one of our courses.
Oh, and this blog is dedicated to Terry, for reminding me that I have teeth.
Image: "Pez Kids Breakfast Cereal Prizes," courtesy of Mike Mozart