Question

use the tools to circumscribe a circle about the triangle.

Explanation:

Step1: Find perpendicular bisectors

To circumscribe a circle around a triangle, we first need to find the perpendicular bisectors of two sides of the triangle. The perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two equal parts. We can use the straight - edge tool (the ruler - like tool) to draw the perpendicular bisectors of two sides of the triangle. For example, take two sides of the triangle, say side \(AB\) and side \(BC\). For side \(AB\), we can find the mid - point (by measuring or using geometric construction) and then draw a line perpendicular to \(AB\) at its mid - point. Do the same for side \(BC\).

Step2: Find the circumcenter

The point where the two perpendicular bisectors intersect is called the circumcenter (\(O\)) of the triangle. This point is equidistant from all three vertices of the triangle.

Step3: Determine the radius

The distance from the circumcenter \(O\) to any of the vertices of the triangle (e.g., \(OA\), \(OB\), or \(OC\)) is the radius (\(r\)) of the circumscribed circle.

Step4: Draw the circle

Use the compass tool. Place the tip of the compass at the circumcenter \(O\) and set the radius to the distance from \(O\) to a vertex (e.g., \(OA\)). Then, draw the circle. This circle will pass through all three vertices of the triangle, thus circumscribing the triangle.

Answer:

To circumscribe a circle about the triangle, follow these steps:

1. Use the straight - edge to draw the perpendicular bisectors of two sides of the triangle.
2. Find the intersection point of these two perpendicular bisectors (the circumcenter).
3. Use the compass, with the circumcenter as the center and the distance from the circumcenter to a vertex as the radius, to draw the circle. The circle will pass through all three vertices of the triangle.