Difference between revisions of "1956 AHSME Problems/Problem 43"

Latest revision as of 16:29, 19 September 2022

The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is:

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$

We can write all possible triangles adding up to 12 or less $(2, 4, 5)=11$ $(3, 4, 5)=12$ $(2, 3, 4)=9$

This leaves $\boxed{\textbf{(C)} \quad 3}$ scalene triangles.

-coolmath34

-rubslul

(If you see any cases I missed out, edit them in.)

~Lucas

1956 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 42	Followed by Problem 44
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

@@ Line 6: / Line 6: @@
 ==Solution==
-We can write all possible triangles starting with a minimum side length of <math>1.</math>
+We can write all possible triangles adding up to 12 or less
-<cmath>(1, 6, 6)</cmath>
+<cmath>(2, 4, 5)=11</cmath>
-<cmath>(2, 5, 6)</cmath>
+<cmath>(3, 4, 5)=12</cmath>
-<cmath>(3, 4, 6), (3, 5, 7)</cmath>
+<cmath>(2, 3, 4)=9</cmath>
-The first triple <math>(1, 6, 6)</math> is not scalene, because two of the sides are equal. This leaves <math>\boxed{\textbf{(C)} \quad 3}</math> scalene triangles.
+This leaves <math>\boxed{\textbf{(C)} \quad 3}</math> scalene triangles.
 -coolmath34
+-rubslul
 (If you see any cases I missed out, edit them in.)
+==Video Solution==
+https://youtu.be/LYFaYLiLTXE
+~Lucas
 ==See Also==
 {{AHSME 50p box|year=1956|num-b=42|num-a=44}}
 {{MAA Notice}}