1. **Problem 4:** Find the length of side $DF$ in triangle $DEF$ or state what additional information is needed. 2. Given triangle $DEF$ with angles $D=56^\circ$, $E=39^\circ$, and side $EF=6$. To find $DF$, we need either the length of another side or the measure of angle $F$. 3. Calculate angle $F$ using the triangle angle sum rule: $$ F = 180^\circ - D - E = 180^\circ - 56^\circ - 39^\circ = 85^\circ $$ 4. Use the Law of Sines to find $DF$: $$ \frac{DF}{\sin E} = \frac{EF}{\sin D} \implies DF = \frac{EF \cdot \sin E}{\sin D} $$ 5. Substitute known values: $$ DF = \frac{6 \cdot \sin 39^\circ}{\sin 56^\circ} $$ 6. Calculate sine values: $$ \sin 39^\circ \approx 0.6293, \quad \sin 56^\circ \approx 0.8290 $$ 7. Compute $DF$: $$ DF = \frac{6 \times 0.6293}{0.8290} \approx \frac{3.7758}{0.8290} \approx 4.55 $$ --- 8. **Problem 5a:** Dilate triangle $ABC$ with center $P$ and scale factor $\frac{3}{2} = 1.5$. 9. The dilation multiplies all side lengths by $1.5$: - $AB' = 15 \times 1.5 = 22.5$ - $BC' = 20 \times 1.5 = 30$ - $AC' = 24 \times 1.5 = 36$ 10. **Problem 5b:** Properties of dilations tell us that angles remain unchanged. 11. Therefore, angle $B$ after dilation is still $56^\circ$. **Final answers:** - Length $DF \approx 4.55$ - Dilated sides: $AB' = 22.5$, $BC' = 30$, $AC' = 36$ - Angle $B$ remains $56^\circ$ after dilation.