1. **Problem Statement:** We have a right triangle $\triangle ACE$ with sides $AC=12$, $CE=16$, and $EA=20$. Points $B$, $D$, and $F$ lie on sides $AC$, $CE$, and $EA$ respectively, with $AB=3$, $CD=4$, and $EF=5$. We need to find the ratio of the area of $\triangle DEF$ to that of $\triangle BDF$. 2. **Key Idea:** To find the ratio of areas of two triangles sharing some vertices, we can use coordinate geometry or area formulas. Since $\triangle ACE$ is right angled at $C$, we place it on coordinate axes for convenience. 3. **Set Coordinates:** Place $C$ at origin $(0,0)$. - Since $AC=12$ and $CE=16$ with right angle at $C$, let $A=(12,0)$ and $E=(0,16)$. 4. **Locate Points $B$, $D$, and $F$:** - $B$ lies on $AC$ with $AB=3$. Since $A=(12,0)$ and $C=(0,0)$, $B$ is 3 units from $A$ towards $C$ along $x$-axis: $B=(12-3,0)=(9,0)$. - $D$ lies on $CE$ with $CD=4$. Since $C=(0,0)$ and $E=(0,16)$, $D$ is 4 units from $C$ towards $E$ along $y$-axis: $D=(0,4)$. - $F$ lies on $EA$ with $EF=5$. Since $E=(0,16)$ and $A=(12,0)$, vector $EA = (12-0,0-16) = (12,-16)$. 5. **Find $F$ coordinates:** - $F$ is 5 units from $E$ towards $A$ along $EA$. - Length of $EA$ is 20, so unit vector from $E$ to $A$ is $(\frac{12}{20}, \frac{-16}{20}) = (0.6, -0.8)$. - Thus, $F = E + 5 \times (0.6, -0.8) = (0 + 3, 16 - 4) = (3, 12)$. 6. **Calculate Areas:** - Use the shoelace formula or determinant formula for area of triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$: $$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ 7. **Area of $\triangle DEF$:** - $D=(0,4)$, $E=(0,16)$, $F=(3,12)$ $$\text{Area}_{DEF} = \frac{1}{2} |0(16-12) + 0(12-4) + 3(4-16)| = \frac{1}{2} |0 + 0 + 3(-12)| = \frac{1}{2} | -36| = 18$$ 8. **Area of $\triangle BDF$:** - $B=(9,0)$, $D=(0,4)$, $F=(3,12)$ $$\text{Area}_{BDF} = \frac{1}{2} |9(4-12) + 0(12-0) + 3(0-4)| = \frac{1}{2} |9(-8) + 0 + 3(-4)| = \frac{1}{2} |-72 -12| = \frac{1}{2} |-84| = 42$$ 9. **Ratio:** $$\frac{\text{Area}_{DEF}}{\text{Area}_{BDF}} = \frac{18}{42} = \frac{3}{7}$$ **Final answer:** (E) $\frac{3}{7}$