IPMAT Indore 2025 · Quantitative Ability · Question 27

Step 1 — picture the path. Staying exactly 1 m away from the triangle at all times traces an outline 1 m outside it. That outline has two kinds of pieces:

three straight segments, each parallel to a side

three circular arcs, one wrapping around each corner

Step 2 — the straight parts. A segment parallel to a side, offset outwards, has exactly the same length as that side. So the straights total

4+6+9=19 m

Step 3 — the corner arcs. This is the elegant bit. Each arc has radius 1, and the three arc angles are the triangle’s exterior angles. The exterior angles of any polygon add up to {360}^{∘} — so the three arcs together form one complete circle of radius 1:

\text{arc total}=2\pi \left( 1 \right)=2\pi m

Step 4 — add.

19+2\pi

Note the side lengths 4, 6, 9 never actually matter beyond their sum.

Answer: Option 2 — 19+2\pi