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? options: 6, 7, 8, 9

Explanation:

Step1: Recall triangle inequality and acute triangle condition

For a triangle with sides \(a\), \(b\), \(c\) (where \(c\) is the longest side), the triangle inequality is \(a + b>c\). For an acute triangle, \(a^{2}+b^{2}>c^{2}\). Here, \(a = x\), \(b = 2x\), \(c = 15\).

First, apply the triangle inequality: \(x + 2x>15\)
\(3x>15\)
\(x > 5\)

Step2: Apply acute triangle condition

\(x^{2}+(2x)^{2}>15^{2}\)
\(x^{2}+4x^{2}>225\)
\(5x^{2}>225\)
\(x^{2}>45\)
\(x>\sqrt{45}\approx6.708\)

Since \(x\) is a whole number and \(x > 6.708\) and \(x>5\), the smallest whole - number value of \(x\) is \(7\).

Answer:

7