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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.

Respuesta :

frika

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

[tex]x<y<4x[/tex]

The perimeter is

[tex]x+y+4x=60\\ \\5x+y=60[/tex]

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

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

[tex]4x=30\\ \\x=7.5\\ \\y=60-5\cdot 7.5=22.5[/tex]

But

[tex]x+y=7.5+22.5=30\ cm[/tex]

This means that such triangle does not exist

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

[tex]y=60-5\cdot 7=25\ cm[/tex]

Since,

[tex]7+25=32>28\\ \\7+28=35>25\\ \\25+28=53>7,[/tex]

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

[tex]y=60-5\cdot 5=35\ cm[/tex]

But this triangle does not exist, because [tex]5+20<35[/tex]

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:  https://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

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