Answer to Yesterday's Problem:
25 triangles. The perimeter will be a + a + b = 99, where a represents the lengths of the equal sides and b represents the base, or the third side. It follows that a + a must be even, since the sum of two odd numbers or the sum of two even numbers will always be even. Therefore, the base b must be an odd number in length to allow an even number plus b to equal 99. Therefore, to test the numbers, the possible lengths of the base must include 1 to 97, inclusive. However, the sum of the two equal lengths must be greater than the length of the base for the three segments to form a triangle. When the base length reaches 51, this situation is no longer possible (24 + 24 = 48). Therefore, the greatest length of a base will be 49 inches. The base includes the odd numbers 1 to 49, inclusive, which will provide 25 triangles. The possibilities are listed below.
25, 25, 49
26, 26, 47
27, 27, 45
28, 28, 43
29, 29, 41
...
49, 49, 1
Today's Problem (5 points):
Find all the times, on the hour, when the minute and hour hands of a clock form a 90 degree angle.
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