Question

multiple choice 4 points
which of the following sets of numbers could represent the lengths of the sides of a
triangle?
2,2,5
4,4,8
5,5,15
3,3,5
multiple choice 4 points
if the sides of a triangle are 3,7, and x, which of the following best describes x?
x < 10
x > 4
4 ≤ x ≤ 10
4 < x < 10
multiple choice 4 points
eva has four sets of straws. the measurements of the straws are given below. which
set of straws could not be used to form a triangle?
set 1: 4cm, 4cm, 7cm
set 2: 2cm, 3cm, 8cm
set 4: 5cm, 12cm, 13cm
set 3: 3cm, 4cm, 5cm

Explanation:

Step1: Triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b|\lt c\lt a + b\).

Step2: Apply to problem 1

• For \(2,2,5\): \(2+2 = 4\lt5\), so it cannot form a triangle.
• For \(4,4,8\): \(4 + 4=8\), so it cannot form a triangle.
• For \(5,5,15\): \(5+5 = 10\lt15\), so it cannot form a triangle.
• For \(3,3,5\): \(3+3=6\gt5\), \(|3 - 3|=0\lt5\), \(|3 - 5| = 2\lt3\), \(|5 - 3|=2\lt3\), so it can form a triangle.

Step3: Apply to problem 2

Given side lengths \(3\) and \(7\). By triangle - inequality: \(|7 - 3|\lt x\lt7 + 3\), i.e., \(4\lt x\lt10\).

Step4: Apply to problem 3

• Set 1: \(4+4 = 8\gt7\), \(|4 - 4|=0\lt7\), \(|4 - 7|=3\lt4\), \(|7 - 4|=3\lt4\), can form a triangle.
• Set 2: \(2+3=5\lt8\), cannot form a triangle.
• Set 3: \(3+4 = 7\gt5\), \(|3 - 4|=1\lt5\), \(|3 - 5|=2\lt4\), \(|4 - 5|=1\lt3\), can form a triangle.
• Set 4: \(5+12 = 17\gt13\), \(|5 - 12|=7\lt13\), \(|5 - 13|=8\lt12\), \(|12 - 13|=1\lt5\), can form a triangle.

Answer:

1. \(3,3,5\)
2. \(4\lt x\lt10\)
3. Set 2: \(2\mathrm{cm},3\mathrm{cm},8\mathrm{cm}\)