Question

The perimeter of the scalene triangle is 60 cm. The length

of the longest side is 4 times that of the shortest side.
Which statements about the possible measures of the
sides are reasonable? Check all that apply.
The value of x can equal 40.
The longest side can equal 30 cm.
The shortest side can equal 7 cm.
The value of x can equal 25.
The shortest side can equal 5.

Answer 1

Explanation:

Perimeter = 60 cm

longest side = 4* shortest side (x)

longest side = 4x

shortest side = x

third (intermediate side=y= 60 -x -4x = 60-5x

The triangle inequality specifies that the sum of the two shorter sides must be greater than the longest side to form a triangle. Hence

x + y > 4x

x + 60-5x > 4x

60 - 4x > 4x

8x < 60

x < 60/8 = 7.5, or

x < 7.5

Therefore to form a triangle, x must be less than 7.5 cm.

Now if we look at the options both 7 and 5 are less than 7.5 cm.

40, 30 and 25 all have a problem because the longest side (4 times longer) will exceed the perimeter of 60.

Now also examine cases where 4x is not the longest side, in which case we need

4x>=y

or

4x >= 60-5x

9x >=60

x >= 6.67

so x=5 will not qualify, because 4x will no longer be the longest side.

The only valid option is x=7 cm

The side lengths for x=7 and x=5 are, respectively,

(7, 25, 28)....