Question

find the measure of $overline{pq}$. answer attempt 1 out of 2

Explanation:

Step1: Apply the triangle - inequality theorem

In a triangle, the length of any side must be less than the sum of the lengths of the other two sides and greater than the absolute value of the difference of the lengths of the other two sides. Let the sides of the triangle be \(a = 9\), \(b = 16\), and \(c=\overline{PQ}\). Then \(|a - b|\lt c\lt a + b\).

Step2: Calculate the range

First, calculate \(|9 - 16|=| - 7| = 7\) and \(9+16 = 25\). So \(7\lt\overline{PQ}\lt25\). Without more information (such as angle - measures to use the Law of Cosines), if we assume this is a non - degenerate triangle, and we are looking for a possible value within the range. If the triangle is a non - degenerate triangle and we assume no other special conditions, we note that for a triangle with side lengths \(a = 9\) and \(b = 16\), by the triangle inequality, the length of \(\overline{PQ}\) satisfies \(7\lt\overline{PQ}\lt25\). If we assume this is a right - triangle and we want to find the length using the Pythagorean theorem (assuming \(\overline{PQ}\) is the hypotenuse), then \(\overline{PQ}=\sqrt{9^{2}+16^{2}}=\sqrt{81 + 256}=\sqrt{337}\approx18.36\) (but we are not given that it is a right - triangle). Since no angle information is given, we just state the range from the triangle inequality.

Answer:

The length of \(\overline{PQ}\) satisfies \(7\lt\overline{PQ}\lt25\)