Partition a Segment in a Given Ratio

Idea

To partition a segment in a given ratio means to place a point on the segment so that the two smaller parts have lengths in the stated proportion. If the ratio is (m:n), then the whole segment is thought of as (m+n) equal parts.

Method

1. Identify the total ratio parts. Add the two numbers in the ratio.
2. Find one part. Divide the whole segment length by (m+n).
3. Measure each section.
   • The first part has length (m) times one unit part.
   • The second part has length (n) times one unit part.
4. Write the exact answer. If the answer involves fractions or radicals, leave it in exact simplified form.

Check

Your two segment lengths should add back to the full segment. Also, their ratio should simplify to the required (m:n).

Tip

If the point is closer to one endpoint, that endpoint usually corresponds to the smaller number in the ratio, because that side has the shorter length.