Answer :
To find the circumcenter of a triangle ABC with side lengths [tex]\(AB = 5\)[/tex] cm, [tex]\(BC = 6\)[/tex] cm, and [tex]\(AC = 7\)[/tex] cm, we'll carry out the following steps:
### Step 1: Construct the Triangle
1. Draw [tex]\(AB\)[/tex]
- Draw a line segment AB of length 5 cm.
2. Set the Compass for [tex]\(AC\)[/tex]
- Place the compass at point A and set the radius to 7 cm (AC).
- Draw an arc.
3. Set the Compass for [tex]\(BC\)[/tex]
- Place the compass at point B and set the radius to 6 cm (BC).
- Draw another arc to intersect the previous arc.
4. Identify Point [tex]\(C\)[/tex]
- Mark the intersection of the two arcs as point C.
- Connect points A and C and points B and C to complete triangle ABC.
### Step 2: Draw the Perpendicular Bisectors of the Sides [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], and [tex]\(AC\)[/tex]
1. Perpendicular Bisector of side [tex]\(AB\)[/tex]
- Find the midpoint of [tex]\(AB\)[/tex]. Since [tex]\(AB = 5\)[/tex] cm, the midpoint [tex]\(M\)[/tex] will be at [tex]\((2.5, 0)\)[/tex].
- Using a compass, place the needle at [tex]\(A\)[/tex] and draw an arc above and below [tex]\(AB\)[/tex] with radius more than half the length of [tex]\(AB\)[/tex].
- Repeat the same arc drawing with the needle at [tex]\(B\)[/tex].
- Draw a straight line through the intersection points of these arcs. This is the perpendicular bisector of [tex]\(AB\)[/tex].
2. Perpendicular Bisector of side [tex]\(BC\)[/tex]
- Find the midpoint of [tex]\(BC\)[/tex].
- Place the needle of the compass at [tex]\(B\)[/tex] with radius more than half the length of [tex]\(BC\)[/tex] and draw arcs above and below [tex]\(BC\)[/tex].
- Repeat the same with the needle at [tex]\(C\)[/tex].
- Draw a straight line through intersection points. This is the perpendicular bisector of [tex]\(BC\)[/tex].
3. Perpendicular Bisector of side [tex]\(AC\)[/tex]
- Find the midpoint of [tex]\(AC\)[/tex].
- Place the needle of the compass at [tex]\(A\)[/tex] with radius more than half the length of [tex]\(AC\)[/tex] and draw arcs above and below [tex]\(AC\)[/tex].
- Repeat the same with the needle at [tex]\(C\)[/tex].
- Draw a straight line through intersection points. This is the perpendicular bisector of [tex]\(AC\)[/tex].
### Step 3: Identify the Circumcenter
- The circumcenter of triangle ABC is found at the point where the three perpendicular bisectors intersect.
### Conclusion
The circumcenter is equidistant from all vertices of the triangle and lies at the point where all three perpendicular bisectors intersect.
### Step 1: Construct the Triangle
1. Draw [tex]\(AB\)[/tex]
- Draw a line segment AB of length 5 cm.
2. Set the Compass for [tex]\(AC\)[/tex]
- Place the compass at point A and set the radius to 7 cm (AC).
- Draw an arc.
3. Set the Compass for [tex]\(BC\)[/tex]
- Place the compass at point B and set the radius to 6 cm (BC).
- Draw another arc to intersect the previous arc.
4. Identify Point [tex]\(C\)[/tex]
- Mark the intersection of the two arcs as point C.
- Connect points A and C and points B and C to complete triangle ABC.
### Step 2: Draw the Perpendicular Bisectors of the Sides [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], and [tex]\(AC\)[/tex]
1. Perpendicular Bisector of side [tex]\(AB\)[/tex]
- Find the midpoint of [tex]\(AB\)[/tex]. Since [tex]\(AB = 5\)[/tex] cm, the midpoint [tex]\(M\)[/tex] will be at [tex]\((2.5, 0)\)[/tex].
- Using a compass, place the needle at [tex]\(A\)[/tex] and draw an arc above and below [tex]\(AB\)[/tex] with radius more than half the length of [tex]\(AB\)[/tex].
- Repeat the same arc drawing with the needle at [tex]\(B\)[/tex].
- Draw a straight line through the intersection points of these arcs. This is the perpendicular bisector of [tex]\(AB\)[/tex].
2. Perpendicular Bisector of side [tex]\(BC\)[/tex]
- Find the midpoint of [tex]\(BC\)[/tex].
- Place the needle of the compass at [tex]\(B\)[/tex] with radius more than half the length of [tex]\(BC\)[/tex] and draw arcs above and below [tex]\(BC\)[/tex].
- Repeat the same with the needle at [tex]\(C\)[/tex].
- Draw a straight line through intersection points. This is the perpendicular bisector of [tex]\(BC\)[/tex].
3. Perpendicular Bisector of side [tex]\(AC\)[/tex]
- Find the midpoint of [tex]\(AC\)[/tex].
- Place the needle of the compass at [tex]\(A\)[/tex] with radius more than half the length of [tex]\(AC\)[/tex] and draw arcs above and below [tex]\(AC\)[/tex].
- Repeat the same with the needle at [tex]\(C\)[/tex].
- Draw a straight line through intersection points. This is the perpendicular bisector of [tex]\(AC\)[/tex].
### Step 3: Identify the Circumcenter
- The circumcenter of triangle ABC is found at the point where the three perpendicular bisectors intersect.
### Conclusion
The circumcenter is equidistant from all vertices of the triangle and lies at the point where all three perpendicular bisectors intersect.
