Question

Use the image to answer the 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) Responses 42 inches and 50 inches 42 inches and 50 inches 28 inches and 64 inches 28 inches and 64 inches 36 inches and 92 inches 36 inches and 92 inches 48 inches and 76 inches

Answer 1

To determine which pairs of lengths can be the other two sides of a triangle with the third side being $x$ (as indicated by the range on the number line), we can apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given that $x$ is the length of the third side, our possible pairs of side lengths can be denoted as $a$ and $b$. The inequalities we need to satisfy for side lengths $a$ and $b$ with the third side $x$ are:

1. $a+b>x$
2. $a+x>b$
3. $b+x>a$

For each pair of lengths you've provided, we would check whether they satisfy these inequalities within the bounds of $x$ as indicated on the number line.

However, since the numerical range for $x$ isn't given in your question, let’s analyze the options using hypothetical limits:

1. 42 inches and 50 inches: $a=42$, $b=50$

   • Check $a+b>x$: $42+50=92$ should be greater than $x$
   • Check $a+x>b$: $42+x>50$ (x must be > 8)
   • Check $b+x>a$: $50+x>42$ (x must be > -8)

2. 28 inches and 64 inches: $a=28$, $b=64$

   • $28+64=92>x$
   • $28+x>64$ (x must be > 36)
   • $64+x>28$ (x must be > -36)

3. 36 inches and 92 inches: $a=36$, $b=92$

   • $36+92=128>x$
   • $36+x>92$ (x must be > 56)
   • $92+x>36$ (x must be > -56)

4. 48 inches and 76 inches: $a=48$, $b=76$

   • $48+76=124>x$
   • $48+x>76$ (x must be > 28)
   • $76+x>48$ (x must be > -28)

Determine the values of $x$ based on the constraints and compare those with the given ranges. As I don't have the specific range from the number line, you will need to analyze if any of these pairs fall within your defined $x$ range, using the inequalities provided.

If you can provide the specific range from the number line, I can help you identify the correct option directly!