Question

use what you know about the triangle inequality theorem to find the missing potential third length of the triangle.
a. triangle with sides y, 10, 17 minimum value: blank maximum value: blank
b. triangle with sides 17, 15, y minimum value: blank maximum value: blank

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let the sides of the triangle be \(a\), \(b\), and \(c\). Then \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Solve for part a

Let the sides of the triangle be \(a = 10\), \(b = 17\), and \(c=y\).

• For the minimum value of \(y\): \(|17 - 10|7\). The minimum value of \(y\) (when considering non - degenerate triangles) is \(8\) (since side lengths are positive integers).
• For the maximum value of \(y\): \(y<10 + 17\), so \(y<27\). The maximum value of \(y\) is \(26\).

Step3: Solve for part b

Let the sides of the triangle be \(a = 15\), \(b = 17\), and \(c = y\).

• For the minimum value of \(y\): \(|17 - 15|2\). The minimum value of \(y\) (non - degenerate triangle) is \(3\).
• For the maximum value of \(y\): \(y<15 + 17\), so \(y<32\). The maximum value of \(y\) is \(31\).

Answer:

a. Minimum Value: \(8\), Maximum Value: \(26\)
b. Minimum Value: \(3\), Maximum Value: \(31\)