Difference between revisions of "2002 AMC 12P Problems/Problem 16"
(→Solution) |
(→Solution) |
||
| Line 15: | Line 15: | ||
== Solution == | == Solution == | ||
| − | Let <math>a, b,</math> and <math>c</math> denote the bases of altitudes <math>12, 15,</math> and <math>20,</math> respectively. Since they are all altitudes and bases of the same triangle, they have the same area, so <math>\frac{12a}{2}=\frac{15b}{2}=\frac{20c}{2}.</math> Multiplying by <math>2</math>, we get <math>12a=15b=20c.</math> Notice that a simple solution to the equation is if all of them equal <math>12 \cdot 15 \cdot 20.</math> That means <math>a=15 \cdot 20 | + | Let <math>a, b,</math> and <math>c</math> denote the bases of altitudes <math>12, 15,</math> and <math>20,</math> respectively. Since they are all altitudes and bases of the same triangle, they have the same area, so <math>\frac{12a}{2}=\frac{15b}{2}=\frac{20c}{2}.</math> Multiplying by <math>2</math>, we get <math>12a=15b=20c.</math> Notice that a simple solution to the equation is if all of them equal <math>12 \cdot 15 \cdot 20.</math> That means <math>a=15 \cdot 20, b=12 \cdot 20,</math> and <math>c=12 \cdot 15.</math> Simplifying our solution to check for Pythagorean triples we see that this is just a Pythagorean triple, namely a <math>3-4-5</math> triangle. Since the other two angles of a right triangle must be acute, the right angle must be the greatest angle. Therefore, our answer is |
== See also == | == See also == | ||
{{AMC12 box|year=2002|ab=P|num-b=15|num-a=17}} | {{AMC12 box|year=2002|ab=P|num-b=15|num-a=17}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 00:54, 31 December 2023
Problem
The altitudes of a triangle are 
and 
The largest angle in this triangle is
Solution
Let 
and 
denote the bases of altitudes and 
respectively. Since they are all altitudes and bases of the same triangle, they have the same area, so 
Multiplying by 
, we get 
Notice that a simple solution to the equation is if all of them equal 
That means 
and 
Simplifying our solution to check for Pythagorean triples we see that this is just a Pythagorean triple, namely a 
triangle. Since the other two angles of a right triangle must be acute, the right angle must be the greatest angle. Therefore, our answer is
See also
| 2002 AMC 12P (Problems • Answer Key • Resources) | |
| Preceded by Problem 15 |
Followed by Problem 17 |
| 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 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
