Leaving Cert Mathematics Proofs and Constructions: Let triangle $ABC$ be a triangle

Question

Let triangle $ABC$ be a triangle. Prove that if a line $l$ is parallel to $BC$ and cuts $[AB]$ in the ratio $s : t$, where $s, t \in \mathbb{N}$, then it also cuts $[AC]$ in the same ratio.

**Diagram:**

(Insert diagram showing triangle ABC with line XY parallel to BC)

**Given:**
A triangle $ABC$ and a line $XY$ parallel to $BC$ which cuts $AB$ in the ratio $s : t$ where $s, t \in \mathbb{N}$.

**To Prove:**
$[AY] : [YC] = s : t$.

**Construction:**
Divide $[AB]$ into $s + t$ equal parts, $s$ of them lying along $[AX]$ and $t$ of them lying along $[XB]$.

Let triangle $ABC$ be a triangle - Leaving Cert Mathematics - Question 6(a) - 2018

Worked Solution & Example Answer:Let triangle $ABC$ be a triangle - Leaving Cert Mathematics - Question 6(a) - 2018

Divide $[AB]$ into $s + t$ equal parts

Prove the ratio of segments in $AC$

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