Question

value: 5
two sides of a triangle have the measures 12 and 10. find the range of possible measures for the third side.

a. 2 < x < 22
b. 10 < x < 12
c. 2 < x < 12
d. 10 < x < 22

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side. Let the sides of the triangle be \(a = 12\), \(b = 10\), and the third - side be \(x\).

Step2: Calculate the upper - bound

\(a + b>x\), so \(12 + 10>x\), which simplifies to \(x<22\).

Step3: Calculate the lower - bound

\(\vert a - b\vert

Answer:

A. \(2 < x < 22\)