the shorter sides of an acute triangle are x cm and 2x cm. the longest side of the triangle is 15 cm. what…
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
Answer
Explanation:
Step1: Apply the acute - triangle inequality
For an acute triangle with sides (a), (b), (c) ((c) is the longest side), (a^{2}+b^{2}>c^{2}). Here (a = x), (b = 2x), (c = 15). So (x^{2}+(2x)^{2}>15^{2}).
Step2: Simplify the inequality
Expand the left - hand side: (x^{2}+4x^{2}>225), which is (5x^{2}>225). Divide both sides by 5: (x^{2}>45).
Step3: Solve for (x)
Take the square root of both sides. Since (x>0), (x>\sqrt{45}\approx6.71).
Answer:
7