The tri county schools are participating in a renaissance faire. The team from Appleton High school has designed a pennant to fly over its team tent.The pennant is in the shape of a triangle. The longest side is 3 inches longer than half the middle length side. If the team captain bought 72 of satin edging for the banner , and there were 4 inches left over after the pennant was made , what are the dimensions of the pennant?
let the shortest side be a, let the middle side be b, then the longest side is
b/2 + 3.
so the perimeter is a + b + b/2 + 3
then a+b+b/2+3 + 4 = 72
a = 65 - 3b/2
but b/2+3 is said to be the longest side, so
b/2 + 3 > b
b + 6 > 2b
b < 6
let b = 2, then a = 62 , other side is 4 , no such triangle is possible
let b = 4, then a = 59, other side is 5, no such triangle
let b = 30, then a = 20, other side is 18
This does form a triangle, but 18 comes from the definition of the "longest" side. This is a contradiction. The same would be true for all b > 6
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To find the dimensions of the pennant, let's assign variables to the different sides of the triangle. Let:
- x be the length of the middle side (in inches).
- 3 + (1/2)x be the length of the longest side (in inches).
We're given that the team captain bought 72 inches of satin edging for the banner, with 4 inches left over. So, we can set up the equation:
x + (x/2) + (3 + (1/2)x) = 72 + 4
Simplifying this equation:
2x + (3 + (1/2)x) = 76
Combining like terms:
3x + 3 = 76
Subtracting 3 from both sides:
3x = 73
Dividing both sides by 3:
x = 73/3
Therefore, the length of the middle side (x) is approximately 24.33 inches.
Now, we can find the length of the longest side by substituting x into the equation:
Longest side = 3 + (1/2)x
Longest side = 3 + (1/2)(73/3)
Longest side = 3 + 73/6
Longest side = 18.17 inches
So, the dimensions of the pennant are approximately:
- Middle side: 24.33 inches
- Longest side: 18.17 inches.