Question

the shorter sides of an acute triangle are x cm and 2x cm. the longest side of the triangle is 15 cm. what is the smallest possible whole - number value of x? 6 7 8 9

Explanation:

Step1: Apply the triangle - inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\). Here, \(a=x\), \(b = 2x\), and \(c = 15\). So \(x+2x>15\), which simplifies to \(3x>15\) or \(x > 5\).

Step2: Apply the Pythagorean - like inequality for acute - angled triangles

In an acute - angled triangle, \(a^{2}+b^{2}>c^{2}\). Substituting \(a=x\), \(b = 2x\), and \(c = 15\), we get \(x^{2}+(2x)^{2}>15^{2}\). Expanding, \(x^{2}+4x^{2}>225\), \(5x^{2}>225\), \(x^{2}>45\), \(x>\sqrt{45}\approx6.71\).

Answer:

7