In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
One side = L units.
2nd side = 3L units.
3rd side = 15 units.
3L < 15
L < 5
L = 4 = Largest possible integer.
3L = 3*4 = 12.
P = 4 + 12 + 15 = 31 = Max. possible
perimeter.
for sides a<b<c
c-b < a < c+b
So, if 3L is the longest side
15-L < 3L < L+15
15 < 4L < 2L+15
L > 3.75
L < 7.5
So, the extreme is 7,15,21 for a perimeter of 43
thank you
To find the greatest possible perimeter of the triangle, we need to determine the length of the other two sides.
Let's call the second side "x." According to the information given, the first side is three times as long as the second side. So, the length of the first side is 3x.
To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. By using this information, we can set up an inequality to find the possible range of values for x:
x + 3x > 15
Combining like terms, we have:
4x > 15
To solve for x, divide both sides of the inequality by 4:
x > 15 / 4
x > 3.75
Since x must be an integer, the smallest integer greater than 3.75 is 4. Therefore, the second side of the triangle can be 4 or any integer greater than 4.
Now we can find the greatest possible perimeter by considering the maximum values for the two sides. Let's assume the second side is 4. In that case, the first side would be 3 times 4, which is 12.
The perimeter of the triangle is the sum of all three sides: 4 + 12 + 15 = 31.
Thus, the greatest possible perimeter of the triangle is 31 units.