Here are the solutions to the problems:
-
Construct triangle ABQ equal in area to triangle ABC.
To construct a triangle ABQ with the same area as triangle ABC, sharing the same base AB, the vertex Q must lie on a line parallel to AB and passing through C.
- Step 1: Draw a line through point C that is parallel to the base AB.
- Step 2: Choose any point Q on this parallel line.
- Step 3: Connect points A to Q and B to Q.
- Triangle ABQ will have the same base AB and the same perpendicular height (distance between parallel lines) as triangle ABC, thus having an equal area.
-
**Evaluate $\int (