Experts Please Help with this PS

srini1988 wrote:If the sides of a triangle have lengths x,y, and z, x+y=30 and y+z=20. then which of the following could be the perimeter of the triangle.

I.28
II.36
III.42

A)I only
B)II only
C)I and II only
D)I and III only
E)I,II and III

First of all, the perimeter CANNOT equal 28
We know this because we're told that x + y = 30, which means the sum of two sides is 30
In order for the perimeter (x+y+z) to equal 28, side z would have to have length -2, which makes no sense.
ELIMINATE A, C, D, and E

Answer: B

On test day, I wouldn't spend any more time on this question.
However, let's keep going. . .

Next, we can show that the perimeter CANNOT equal 42
IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
DIFFERENCE between A and B < length of third side < SUM of A and B
We're told that y + z = 20, which means the sum of sides y and z is 20
The above rule tells us that the third side (side x) must be LESS THAN 20
If x is less than 20, and y+z = 20, it's impossible for the perimeter (x+y+z) to equal 42


Finally, the perimeter (x+y+z) CAN equal 36
If x = 16 y = 14, and z = 6, then all of the conditions are met, AND the perimeter is 36

srini1988 wrote:If the sides of a triangle have lengths x,y, and z, x+y=30 and y+z=20. then which of the following could be the perimeter of the triangle.

I.28
II.36
III.42

A)I only
B)II only
C)I and II only
D)I and III only
E)I,II and III

If we add the two equations, we have x + 2y + z = 50. Subtracting x + y + z (i.e, the perimeter of the triangle) from this, we have y = 50 - (x + y + z). Now let’s check the numbers in the given Roman numerals.

I. 28

If the perimeter is 28, then y = 50 - 28 = 22. However, it’s not possible for y + z = 20 (since z would have to be -2). Therefore, 28 can’t be the perimeter.

II. 36

If the perimeter is 36, then y = 50 - 36 = 14. In this case, x + 14 = 30 → x = 16 and 14 + z = 20 → z = 6. So we have x = 16, y = 14 and z = 6. We can see that these 3 numbers can be the side lengths of a triangle.

III. 42

If the perimeter is 42, then y = 50 - 42 = 8. In this case, x + 8 = 30 → x = 22 and 8 + z = 20 → z = 12. So we have x = 22, y = 8 and z = 12. However, these 3 numbers can’t be the side lengths of a triangle since 8 + 12 is not greater than 22.

Answer: B