1. **State the problem:** We have a triangle with one angle measuring 36° and two sides labeled as 2t and 4t. We need to find the value of $t$. 2. **Understand the problem:** Since the problem gives an angle and two sides, and the sides are expressed in terms of $t$, we can use the properties of triangles to find $t$. However, the problem does not specify which sides correspond to which angles or if the triangle is right-angled. Assuming the triangle is valid and the sides correspond to the given angle, we can use the triangle angle sum property or the Law of Sines if more information was given. 3. **Assuming the triangle is isosceles or the angle is between the two sides:** If the angle 36° is between the sides 2t and 4t, we can use the Law of Cosines to find the third side or relate the sides. 4. **Law of Cosines formula:** $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $C$ is the angle between sides $a$ and $b$, and $c$ is the side opposite $C$. 5. **Since the problem does not provide the third side, we cannot apply Law of Cosines directly.** 6. **Alternatively, if the triangle is right-angled and 36° is one of the angles, then the sides 2t and 4t could be legs or hypotenuse.** 7. **Using the triangle angle sum property:** The sum of angles in a triangle is 180°. 8. **If the triangle is right-angled, one angle is 90°, and the other two angles sum to 90°. Given one angle is 36°, the other angle is $90° - 36° = 54°$.** 9. **Using the ratio of sides in a right triangle with angles 36° and 54°:** The sides opposite these angles are in ratio: $$\sin(36°) : \sin(54°)$$ 10. **Calculate the ratio:** $$\sin(36°) \approx 0.5878$$ $$\sin(54°) \approx 0.8090$$ 11. **Given sides 2t and 4t correspond to these opposite sides, set up the ratio:** $$\frac{2t}{4t} = \frac{\sin(36°)}{\sin(54°)}$$ 12. **Simplify the left side:** $$\frac{2t}{4t} = \frac{2}{4} = \frac{1}{2}$$ 13. **Set the equation:** $$\frac{1}{2} = \frac{0.5878}{0.8090}$$ 14. **Calculate the right side:** $$\frac{0.5878}{0.8090} \approx 0.7265$$ 15. **Since $\frac{1}{2} \neq 0.7265$, the assumption that sides 2t and 4t correspond to these angles is incorrect.** 16. **Without additional information about the triangle, such as which sides correspond to which angles or if the triangle is right-angled, the value of $t$ cannot be determined uniquely.** **Final answer:** Insufficient information to determine $t$.