Let's break down the steps to solve this problem in detail:
### Part a
Constructing a Triangle:
1. Draw a line segment AB of length 9 cm.
2. With A as the center and radius 5 cm, draw an arc.
3. With B as the center and radius 7 cm, draw another arc that intersects the first arc at point C.
4. Connect A to C and B to C to form the triangle ABC with sides 5 cm (AC), 7 cm (BC), and 9 cm (AB).
### Part b
Constructing the Perpendicular Bisector of Each Side:
1. For side AB:
- Find the midpoint M of AB by measuring 4.5 cm from point A along the line segment AB.
- Draw a line perpendicular to AB at M.
2. For side AC:
- Find the midpoint N of AC by measuring 2.5 cm from point A along the line segment AC.
- Draw a line perpendicular to AC at N.
3. For side BC:
- Find the midpoint P of BC by measuring 3.5 cm from point B along the line segment BC.
- Draw a line perpendicular to BC at P.
What do you notice?
- The perpendicular bisectors of the sides of the triangle intersect at a single point inside the triangle. This point is called the circumcenter.
### Part c
Drawing a Circle with its Centre at the Circumcenter:
1. Use the point of intersection of the perpendicular bisectors (the circumcenter) as the center.
2. Set your compass to the distance from the circumcenter to any vertex of the triangle (this distance will be the same for each vertex).
3. Draw a circle with this radius. This circle should pass through vertices A, B, and C.
### Part d
Constructing a Different Triangle:
1. Repeat step a to construct another triangle with different side lengths (e.g., sides of 6 cm, 8 cm, and 10 cm).
2. Repeat step b to construct the perpendicular bisectors of each side of the new triangle.
- Find the midpoints of each side.
- Draw lines perpendicular to each side at its midpoint.
3. Once again, observe the point where all three perpendicular bisectors intersect. This point is the new triangle's circumcenter.
4. Repeat step c to draw the circumcircle for the new triangle.
- Use the intersecting point as the center.
- Draw a circle passing through each vertex.
What do you notice?
- In any triangle, the perpendicular bisectors of the sides intersect at a unique point called the circumcenter.
- The circle drawn with the circumcenter as the center will pass through all three vertices of the triangle, forming the circumcircle.
This process demonstrates that the property of the perpendicular bisectors intersecting at the circumcenter, which is equidistant from all vertices, holds true for any triangle.
