Question

The perimeter of the scalene triangle is 60 cm. The length of the longest side is 4 times that of the shortest side.

Answer 1

The longest side of this scalene triangle with a perimeter of 60 cm can equal 30 cm or the shortest side can equal 7 cm.

Further Explanation

We can use the variables x, y and z to represent the shortest (x), medium (y) and longest (z) sides.  The perimeter of a triangle is found by adding together all of the sides; this gives us the equation

x + y + z = 60

We know that the longest side, z, is equal to 4 times the length of the shortest side, x.  This means that z = 4x; we can now write our equation as

x + y + 4x = 60

Combining like terms, we have

5x + y = 60

1.  Checking all of the possible options, we first determine if x can equal 40:

• 5(40) + y = 60
• 200 + y = 60

This would give us a negative side length, which is impossible.

2.  Let the longest side be 30 cm.  This means that the shortest side is 1/4 of that; 30÷4 = 7.5.  Using 7.5 for x,

• 5(7.5)+y = 60
• 37.5 + y = 60
• 37.5 + y - 37.5 = 60-37.5
• y = 22.5

This is within the range of acceptable side lengths, since it is between the smallest (7.5) and the largest (30).

3.  Let the shortest side be 7 cm.  This means x = 7:

• 5(7)+y = 60
• 35+y = 60
• 35+y-35 = 60-35
• y = 25

This is between the longest side, 7 cm, and the longest side, 4(7) = 28 cm.  This is acceptable.

4.  Let the value of x be 25:

• 5(25)+y = 60
• 125+y = 60

This will give us a negative value for the medium side, which is impossible.

5.  Let the shortest side be 5 cm.  This means x = 5:

• 5(5)+y = 60
• 25+y = 60
• 25+y-25 = 60-25
• y = 35

This means the medium value, 35, would be greater than the longest side, 20; this is incorrect.

This means the correct options are that the longest side can be 30 cm and the shortest side can be 7 cm.

Learn More

Learn more about perimeter:  brainly.com/question/12498514

Learn more about equations:  https://brainly.in/question/6788771

Keywords:  perimeter of scalene triangle, finding side lengths of scalene triangles, finding perimeter

Answer 2

Answer:

Only option C: The shortest side can equal 7 cm.

Step-by-step explanation:

Let the length of the shortest side be x cm, then the length of the longest side is 4x cm. Let the length of the middle side be y cm. Note that

$x$

The perimeter is

$x+y+4x=60\\ \\5x+y=60$

A. The value x cannot be 40 cm, because then y is negative

B. If the longest side is 30 cm long, then

$4x=30\\ \\x=7.5\\ \\y=60-5\cdot 7.5=22.5$

But

$x+y=7.5+22.5=30\ cm$

This means that such triangle does not exist

C. If x=7 cm, then 4x=28 cm,

$y=60-5\cdot 7=25\ cm$

Since,

$7+25=32>28\\ \\7+28=35>25\\ \\25+28=53>7,$

such triangle exists and this option is possible

D. If x=25 cm, then y is negative

E. If x=5 cm, then 4x=20 cm and

$y=60-5\cdot 5=35\ cm$

But this triangle does not exist, because $5+20$