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 for any triangle, the length of any side must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. Mathematically, for sides \(a\), \(b\), and \(c\), \(|a - b| < c < a + b\).

Step2: Identify the sides and apply the theorem

Here, the two sides are \(3x\) and \(7x\). First, find the difference: \(|7x - 3x| = |4x| = 4x\) (since \(x\) is a length, \(x>0\), so absolute value is not necessary in the positive sense). Then, find the sum: \(3x + 7x = 10x\).

So, applying the triangle inequality theorem to the side \(h\), we get \(4x < h < 10x\).

Answer:

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