CAT 2025 Slot 2 QA Question 4: In a $ riangle ABC$, points D and E are on the sides BC and AC, respectively. BE and AD intersect at point T such that AD : AT = 4 : 3, and BE : BT = 5 : 4. Point F lies on AC such that DF is paralle

Question

In a $	riangle ABC$, points D and E are on the sides BC and AC, respectively. BE and AD intersect at point T such that AD : AT = 4 : 3, and BE : BT = 5 : 4. Point F lies on AC such that DF is paralle

Accepted Answer

Use an affine coordinate setup, because ratios on straight lines are preserved.

Take $A = (0,1)$, $B = (0,0)$, $C = (1,0)$. Let $D = (d,0)$ and $E = (u,1-u)$.

Since $AT : AD = 3 : 4$, point $T$ divides $AD$ in the ratio $3 : 1$ from $A$.
$T = (\dfrac{3d}{4}, \dfrac{1}{4})$

Since $BT : BE = 4 : 5$

In a △ABC\triangle ABC△ABC, points D and E are on the sides BC and AC, respectively. BE and AD intersect at point T such that AD : AT = 4 : 3, and BE : BT = 5 : 4. Point F lies on AC such that DF is parallel to BE. Then, BD : CD is

In a $\triangle ABC$ , points D and E are on the sides BC and AC, respectively. BE and AD intersect at point T such that AD : AT = 4 : 3, and BE : BT = 5 : 4. Point F lies on AC such that DF is parallel to BE. Then, BD : CD is