1. **Problem 106: Find the length of X in the right triangle**. Given a right triangle with legs 4 and 4, and hypotenuse 7, find the length of segment X. 2. **Formula used:** For a right triangle, by the Pythagorean theorem: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are legs, and $c$ is the hypotenuse. 3. **Apply the formula:** $$4^2 + 4^2 = 7^2$$ Calculate each term: $$16 + 16 = 49$$ $$32 = 49$$ This is not true, so the hypotenuse cannot be 7 if legs are both 4. 4. **Assuming the base segment labeled X is part of the triangle, and the hypotenuse is 7, we can find X using the Pythagorean theorem:** If the legs are 4 and X, and hypotenuse is 7: $$4^2 + X^2 = 7^2$$ $$16 + X^2 = 49$$ Subtract 16 from both sides: $$X^2 = 49 - 16$$ $$X^2 = 33$$ Take the square root: $$X = \sqrt{33} \approx 5.7446$$ 5. **Check options:** 7, 9, 12, 15. None matches exactly, so possibly the problem expects the closest or a different interpretation. Since none matches, the problem might have a typo or different context. --- 6. **Problem 107: Find value of X in the equilateral triangle with angles 60 and 80 degrees and smaller inscribed triangle side X.** Since the triangle is equilateral, all sides are equal and all angles are 60 degrees. Given an angle of 80 degrees, this is not an equilateral triangle, so likely the smaller triangle inside has side X. Without more data, we cannot solve this exactly. **Since the user requested to solve questions 106 and 107, and only 106 has enough data, we provide the solution for 106.** **Final answer for 106:** $$X = \sqrt{33} \approx 5.74$$ No option matches exactly, so none of the given options a)7 b)9 c)12 d)15 is correct based on the Pythagorean theorem.