1. **Stating the problem:** Can you create a two-colored right triangle using two types of triangular jigsaw pieces T5 and T6 without moving the pieces already placed? 2. **Given:** - T5 is a right triangle with sides 1, 1, and hypotenuse $\sqrt{2}$. - T6 is a right triangle with sides 1, $\frac{\sqrt{3}}{2}$, and hypotenuse $\frac{3}{2}$. - Angles involved are 45°, 60°, and 90°. 3. **Goal:** Try to form a larger right triangle composed of T5 and T6 pieces arranged without overlap or moving already placed pieces. 4. **Step 1: Calculate side lengths and angles for T5 and T6.** - For T5, by Pythagoras: hypotenuse $= \sqrt{1^2 + 1^2} = \sqrt{2}$. - For T6, verify hypotenuse: $\left(\frac{\sqrt{3}}{2}\right)^2 + 1^2 = \frac{3}{4} + 1 = \frac{7}{4}$, so hypotenuse $= \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2}$, which is approximately 1.32, not $\frac{3}{2}$, so check the triangle's validity. 5. **Step 2: Analyze if T5 and T6 can fit together to form a larger right triangle.** - Consider combining sides: $1 + \frac{\sqrt{3}}{2}$ and $1 + 1$. - Check if these sums can form sides of a right triangle using Pythagoras. 6. **Step 3: Use Pythagorean theorem to check larger triangle possibility.** - Hypotenuse candidate: $\sqrt{\left(1 + \frac{\sqrt{3}}{2}\right)^2 + (1 + 1)^2}$. - Calculate: $$\left(1 + \frac{\sqrt{3}}{2}\right)^2 = 1 + 2 \times 1 \times \frac{\sqrt{3}}{2} + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 + \sqrt{3} + \frac{3}{4} = \frac{7}{4} + \sqrt{3}$$ $$ (1 + 1)^2 = 2^2 = 4$$ - Sum: $$\frac{7}{4} + \sqrt{3} + 4 = \frac{7}{4} + 4 + \sqrt{3} = \frac{7}{4} + \frac{16}{4} + \sqrt{3} = \frac{23}{4} + \sqrt{3}$$ 7. **Step 4: Since $\sqrt{3} \approx 1.732$, sum is approximately $\frac{23}{4} + 1.732 = 5.75 + 1.732 = 7.482$. - Hypotenuse length $= \sqrt{7.482} \approx 2.735$. 8. **Step 5: Check if this matches any side length of T5 or T6 or their combinations.** - Since no side matches 2.735, forming a perfect larger right triangle without moving pieces is unlikely. 9. **Conclusion:** It is not possible to create a two-colored right triangle from T5 and T6 pieces without moving the pieces already placed. **Final answer:** No, it is not possible to create such a two-colored right triangle without rearranging the pieces.