Question

(HELP ASAP)You have been given a 30 cm pipe cleaner and must make an isosceles triangle with it without any length leftover. Your first bend is 8 cm from one end of the pipe cleaner.

Answer 1

Answer: See below

Explanation:

Isosceles triangles are triangles that have two sides of equal length, with the base being the only length that differs. According to the problem, the pipe cleaner is 30 cm long with an 8 centimetre initial bend. To construct an isosceles triangle, bend the pipe cleaner two more times.

1a. The two combinations that would form an isosceles triangle must have two equal sides and they are:

8 - 8 - 14 triangle

11 - 11 - 8 triangle

1b. The first test that the combinations work is that the sum of the sides is equal to 30 cm as shown below:

`8 + 8 + 14 &\stackrel{?}{=} 30 \text{ cm} \quad & \quad 11 + 11 + 8 &\stackrel{?}{=} 30 \\ 30 &\stackrel{\checkmark}{=} 30 \quad & \quad \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30 &\stackrel{\checkmark}{=} 30`

Another reason stems from the triangle inequality theorem. The theorem states that the sum of two sides is always greater than or at the most equal to the other side of the triangle. To check if this is true for the triangles above look at the sums of the different combinations of the sides:

`8 + 8 &\stackrel{?}{ > } 14 \quad & \quad 11 + 11 &\stackrel{?}{ > } 8 \\ 16 &\stackrel{\checkmark}{ > } 14 \quad & \ \ \ \ \ \ \ \quad 22 &\stackrel{\checkmark}{ > } 8`

Adding the base and the side and comparing it to the other side:

`8 + 14 &\stackrel{?}{ > } 8 \quad & \quad 11 + 8 &\stackrel{?}{ > } 11 \\ 22 &\stackrel{\checkmark}{ > } 8 \quad & \quad \ \ \ \ \ \ \ 19 &\stackrel{\checkmark}{ > } 11`

The sides are all in conformity of the triangle inequality theorem and the lengths all add up to the length of the pipe cleaner, making the combinations work.

2a. No, it is not possible for it to be an equilateral triangle

2b. Since the total length of the pipe cleaner is 30 cm. Because an equilateral triangle has three sides that are the same length, the side length of each of sides has to be = Perimeter/3 = 30/3 = 10 cm

However, the first bend is 8 cm in length.

Therefore, it can't be an equilateral triangle. If it would be, then the side length must be 10 cm without any remainder.