The question asks to construct the perpendicular bisectors on each side of the obtuse triangle $\triangle ABC$. Let's go step by step on how to do this:

Steps to Construct the Perpendicular Bisectors:

1. Identify the Midpoint: For each side of the triangle (AB, BC, and AC), we first need to find the midpoint. To find the midpoint, use a ruler to measure the length of each side, then mark the point exactly halfway along the line.

2. Draw a Perpendicular Line: For each side of the triangle, place the compass needle on the midpoint and draw arcs above and below the line segment (AB, BC, AC). The arcs should intersect the line at two points. Now, connect these intersection points using a straightedge. This line is the perpendicular bisector of the side.

3. Repeat for All Sides: Repeat the process for each of the three sides of the triangle. The perpendicular bisectors will eventually intersect at a single point, known as the circumcenter of the triangle. In an obtuse triangle, the circumcenter lies outside the triangle.

Explanation:

• Perpendicular Bisector: This is a line that divides a line segment into two equal parts at a right angle (90°).
• The circumcenter, where all three perpendicular bisectors meet, is equidistant from all three vertices of the triangle.

If you'd like a more detailed explanation or have questions on a specific step, feel free to ask!

Relative Questions:

1. What are the properties of the perpendicular bisectors in triangles?
2. Why is the circumcenter outside the triangle for obtuse triangles?
3. How do you determine the perpendicular bisector of a line segment geometrically?
4. What is the significance of the circumcenter in relation to the triangle's vertices?
5. Can you construct a perpendicular bisector using only a compass and straightedge?

Tip: When constructing geometric figures like perpendicular bisectors, ensure that your compass is well adjusted to avoid errors in the arcs intersecting correctly.