2011-2012 Mathcounts Right Triangle Stretch

#271: Use Pythagorean Theorem: a² + b² = c²
Also, learn those common triples by heart.

3-4-5 and its derivative: 6-8-10. 9-12-15. 12-15-20, 15-20-25

5-12-13, 10-24-26

8-15-17

7-24-25

9-40-41
12-35-37
20-21-29
4² + 9² = c², c = √ 97 = 9.8

#272: It's one of the triples: 9-12-15

#273: x =  √ 39 = 6.2

#274: In an isosceles triangle, the height will break the base into two equal length and half of the base
is 5 so this is another triple : 5-12-13. x = 12

#275: x is the space diagonal, which can be solved by square root the sum of the square of each of the three
dimensions.
x = √ 18² + 24² + 30² = √1800 = 42.4

#276: Another triple: 9-40-41

#277:  45-45-90 degree angle ratio is 1: 1: √ 2
so the answer is 5√ 2 = 7.1

#278:30-60-90 degree angle ratio is 1: √3: 2
60 degrees is 4√3 so 30 degree is 4 and 90 degree is 4 x 2 = 8

#279: Each side is 8. If you draw a line connecting one vertice to the center x is the side opposite 60 degree.
The length opposite 30 degree is 1/2 of 8 or 4 so the answer is 4√3  = 6.9

#280: 45-45-90 degree angle ratio.
The Length opposite 90 degree is 6 so x is 6/√ 2 = 3√ 2 = 4.2

#281: Pythagorean triple 30-34-16 (derivative of 15-17-8) so y = 16
x is 3 times 34 = 102.

#282: Triangle ACD is similar to triangle ABC. AC/AD = AB/AC = 10/6 = 6/x
x = 3.6. Using the same method y = 4.8