Question

x is the length in inches of the third side of a triangle. the range of all possible values of x is shown on the number line. which of the following options has possible lengths of the other two sides of the triangle?

(1 point)

○ 42 inches and 50 inches

○ 28 inches and 64 inches

○ 36 inches and 92 inches

○ 48 inches and 76 inches

Explanation:

Step1: Recall Triangle Inequality Theorem

For a triangle with sides \(a\), \(b\), and \(x\), the theorem states \(|a - b| < x < a + b\). From the number line, the range of \(x\) is \(36 < x < 92\) (since the open circles are at 36 and 92, meaning \(x\) is strictly between them). So we need \(|a - b| = 36\) and \(a + b = 92\) (or check each option for \(|a - b| < 36\)? Wait, no—wait, the range of \(x\) is \(36 < x < 92\), so the difference of the two sides should be 36 (lower bound) and the sum should be 92 (upper bound). Wait, actually, if the two sides are \(a\) and \(b\), then \(|a - b| < x < a + b\). So the lower bound of \(x\) is \(|a - b|\) and upper bound is \(a + b\). From the number line, lower bound is 36, upper bound is 92. So we need \(|a - b| = 36\) and \(a + b = 92\)? Wait, no, let's check each option:

Step2: Check Option 1: 42 and 50

\(|42 - 50| = 8\), \(42 + 50 = 92\). Wait, \(|a - b| = 8\), \(a + b = 92\). Then \(x\) should be between 8 and 92. But our range is 36 to 92. Not matching.

Step3: Check Option 2: 28 and 64

\(|28 - 64| = 36\), \(28 + 64 = 92\). Ah! \(|a - b| = 36\) (so lower bound of \(x\) is 36) and \(a + b = 92\) (upper bound of \(x\) is 92). So \(x\) would be between 36 and 92, which matches the number line. Let's confirm other options to be sure.

Step4: Check Option 3: 36 and 92

\(|36 - 92| = 56\), \(36 + 92 = 128\). Then \(x\) would be between 56 and 128. Not matching.

Step5: Check Option 4: 48 and 76

\(|48 - 76| = 28\), \(48 + 76 = 124\). Then \(x\) would be between 28 and 124. Not matching.

So the correct option is 28 inches and 64 inches.

Answer:

28 inches and 64 inches (the second option: 28 inches and 64 inches)