Question

11 if two sides of a triangle have lengths of 4 and 10, the third side could be 1) 8 2) 2 3) 16 4) 4 15 sara is building a triangular pen for her pet rabbit. if two of the sides measure 8 feet and 15 feet, the length of the third side could be 1) 13 ft 2) 7 ft 3) 3 ft 4) 23 ft 12 if two sides of a triangle have lengths of \\( \frac{1}{4} \\) and \\( \frac{1}{5} \\), which fraction can not be the length of the third side? 1) \\( \frac{1}{9} \\) 2) \\( \frac{1}{8} \\) 3) \\( \frac{1}{3} \\) 4) \\( \frac{1}{2} \\) 16 the direct distance between city a and city b is 200 miles. the direct distance between city b and city c is 300 miles. which could be the direct distance between city c and city a? 1) 50 miles 2) 350 miles 3) 550 miles 4) 650 miles

Explanation:

Step1: Triangle inequality theorem

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

Step2: Analyze problem 11

Let \(a = 4\) and \(b=10\). Then \(|10 - 4|=6\) and \(10 + 4 = 14\). So \(6\lt c\lt14\).

• For option 1: \(6\lt8\lt14\) (valid)
• For option 2: \(2\lt6\) (invalid)
• For option 3: \(16\gt14\) (invalid)
• For option 4: \(4\lt6\) (invalid)

Step3: Analyze problem 12

Let \(a=\frac{1}{4}\) and \(b = \frac{1}{5}\). Then \(| \frac{1}{4}-\frac{1}{5}|=\frac{5 - 4}{20}=\frac{1}{20}\) and \(\frac{1}{4}+\frac{1}{5}=\frac{5+4}{20}=\frac{9}{20}\). So \(\frac{1}{20}\lt c\lt\frac{9}{20}\approx0.45\)

• For option 1: \(\frac{1}{9}\approx0.11\), \(\frac{1}{20}=0.05\lt\frac{1}{9}\lt\frac{9}{20}\) (valid)
• For option 2: \(\frac{1}{8}=0.125\), \(\frac{1}{20}\lt\frac{1}{8}\lt\frac{9}{20}\) (valid)
• For option 3: \(\frac{1}{3}\approx0.33\), \(\frac{1}{20}\lt\frac{1}{3}\lt\frac{9}{20}\) (valid)
• For option 4: \(\frac{1}{2}=0.5\gt\frac{9}{20}\) (invalid)

Step4: Analyze problem 15

Let \(a = 8\) and \(b = 15\). Then \(|15 - 8|=7\) and \(15+8 = 23\). So \(7\lt c\lt23\)

• For option 1: \(7\lt13\lt23\) (valid)
• For option 2: \(7 = 7\) (invalid)
• For option 3: \(3\lt7\) (invalid)
• For option 4: \(23=23\) (invalid)

Step5: Analyze problem 16

Let \(a = 200\) and \(b = 300\). Then \(|300 - 200|=100\) and \(300+200 = 500\). So \(100\lt c\lt500\)

• For option 1: \(50\lt100\) (invalid)
• For option 2: \(100\lt350\lt500\) (valid)
• For option 3: \(550\gt500\) (invalid)
• For option 4: \(650\gt500\) (invalid)

Answer:

1. 1) \(8\)
2. 4) \(\frac{1}{2}\)
3. 1) \(13\ ft\)
4. 2) \(350\ miles\)