Frank is constructing a triangular pen for his farm animals. He knows that he needs lengths of 13ft and 25ft. What is a possible length for the third side?

Question

10ft
11ft
12ft
13ft

Answer 1

To find a possible length for the third side of the triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side (and vice versa).

Let the lengths of the sides be \(a = 13\) ft, \(b = 25\) ft, and \(c\) be the length of the unknown side. According to the triangle inequality, we have the following conditions:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Substituting \(a\) and \(b\) into these inequalities, we get:

1. \(13 + 25 > c\) → \(c < 38\)
2. \(13 + c > 25\) → \(c > 12\)
3. \(25 + c > 13\) → \(c > -12\) (this condition is always satisfied since \(c\) must be positive)

So combining the useful inequalities, we have:

• \(c > 12\)
• \(c < 38\)

Thus, \(c\) must be greater than 12 but less than 38. The possible lengths given are 10 ft, 11 ft, 12 ft, and 13 ft. The only possible length for the third side that satisfies these inequalities is:

13 ft (since it is greater than 12).

Thus, the answer is 13 ft.

Answer 2

Amari constructed ray ZR, a line segment bisector of XY.
If the length of XY is 10cm, what is the length of XZ?
2.5cm
5cm
10cm
20cm

Answer 3

If \(XY\) is a line segment with a length of 10 cm and \(ZR\) is a line segment bisector of \(XY\), it means that it divides \(XY\) into two equal parts.

The length of \(XZ\) would then be half of the length of \(XY\):

\[ XZ = \frac{XY}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

Therefore, the length of \(XZ\) is 5 cm.