x and y are whole numbers, 10<x<14, and 8<y<14.
The sides of a triangle are 5 cm, x cm, and y cm.
List possible values for x and y.
since any side is less than the sum of the other two sides,
x+y > 5 (duh)
x+5 > y
y+5 > x
lots of possible solutions:
5,11,9
5,13,13
and more
values of x could be 11, 12, or 13
values of y could be 9, 10, 11, 12 or 13
now overthinking the problem:
To have a triangle:
1. x+y > 5
2. x+5 > y ----> x-y > -5 or y-x < 5
3. y+5 > x ----> y-x > -5
from #2 and #2 we can see that
-5 < y-x < 5
so make 3 columns
5_x__y_(y-x)
5 11 9
5 11 10
5 11 11
5 11 12
5 11 13
5 12 9
5 12 10
..
5 12 13
5 13 9
..
5 13 13
in short, the x could be 11, 12 or 13
the y could be 9, 10, 11, 12, or 13
so the number of triangles for sides labeled x , y and 5 is 3(5) or 15
However, triangles 5 , 12, 13 and 5, 13, 12 are the same triangle
there are 3 such duplicated pairs, so
the actual number of triangles possible is 15-3 = 12
To find the possible values for x and y, we need to consider the given conditions:
1. 10 < x < 14: This means x must be greater than 10 and less than 14.
2. 8 < y < 14: This means y must be greater than 8 and less than 14.
We also know that the sides of a triangle must satisfy the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.
In this case, the sides of the triangle are 5 cm, x cm, and y cm. So, to satisfy the triangle inequality, we need to check if the sum of any two sides is greater than the third side:
1. 5 + x > y: The sum of 5 cm and x cm must be greater than y cm.
2. 5 + y > x: The sum of 5 cm and y cm must be greater than x cm.
3. x + y > 5: The sum of x cm and y cm must be greater than 5 cm.
Let's go through the possibilities step by step:
For x = 11:
- 5 + 11 > y: y must be less than 16
- 5 + y > 11: y must be greater than 6
- 11 + y > 5: y must be greater than -6
So, for x = 11, y can range from 7 to 15.
For x = 12:
- 5 + 12 > y: y must be less than 17
- 5 + y > 12: y must be greater than 7
- 12 + y > 5: y must be greater than -7
So, for x = 12, y can range from 8 to 16.
For x = 13:
- 5 + 13 > y: y must be less than 18
- 5 + y > 13: y must be greater than 8
- 13 + y > 5: y must be greater than -8
So, for x = 13, y can range from 9 to 17.
Therefore, the possible combinations of x and y are:
- x = 11, y = 7 to 15
- x = 12, y = 8 to 16
- x = 13, y = 9 to 17
These are the possible values for x and y that satisfy the given conditions.