Exact answer: [tex]x = \frac{\sqrt{601}}{2}\\\\[/tex]
Approximate answer: [tex]x \approx 12.2577[/tex]
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How to find the answer:
Focus on the largest triangle. It appears to be a right triangle. If so, then we can use the pythagorean theorem to get the following:
[tex]a^2 + b^2 = c^2\\\\a^2 + 12^2 = 13^2\\\\a^2 + 144 = 169\\\\a^2 = 169 - 144\\\\a^2 = 25\\\\a = \sqrt{25}\\\\a = 5[/tex]
The horizontal leg of the largest right triangle is 5 units long.
The identical single tickmarks tell us that the horizontal leg is cut into two smaller pieces of equal length.
Each smaller piece is 5/2 = 2.5 units long.
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Now focus on the smaller skinny right triangle with legs a = 2.5, b = 12 and hypotenuse c = x.
Use the pythagorean theorem again to find the hypotenuse x.
[tex]2.5^2 + 12^2 = c^2\\\\6.25 + 144 = x^2\\\\150.25 = x^2\\\\x^2 = 150.25\\\\x = \sqrt{150.25}\\\\x = \sqrt{\frac{601}{4}}\\\\x = \frac{\sqrt{601}}{\sqrt{4}}\\\\x = \frac{\sqrt{601}}{2}\\\\x \approx 12.2577[/tex]
