Question

a triangle has side lengths measuring 3x cm, 7x cm, and h cm. which expression describes the possible values of h, in cm?
4x < h < 10x
10x < h < 4x
h = 4x
h = 10x

Explanation:

Step1: Recall Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For a triangle with sides \(a\), \(b\), and \(c\), we have \(|a - b| < c < a + b\).

Step2: Identify the two given sides

Here, the two given sides are \(3x\) and \(7x\). Let's first find the difference and the sum of these two sides.

Step3: Calculate the difference of the two sides

The difference of \(7x\) and \(3x\) is \(7x - 3x = 4x\).

Step4: Calculate the sum of the two sides

The sum of \(7x\) and \(3x\) is \(7x + 3x = 10x\).

Step5: Apply the Triangle Inequality Theorem to find the range of \(h\)

Using the Triangle Inequality Theorem, the length of the third side \(h\) must be greater than the difference of the other two sides and less than the sum of the other two sides. So, \(|7x - 3x| < h < 7x + 3x\), which simplifies to \(4x < h < 10x\).

Answer:

\(4x < h < 10x\) (the first option)