Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its inscribed circle. Measure the radius of the circle.

Options

A. 1.6 CM
B. 2.8 CM
C. 3.2 CM
D. 2.6 CM

Answer

### Step-by-Step Solution:

1. **Draw the Base of the Triangle:**
- Using a ruler, draw a line segment BC of length 4.5 cm. Label the endpoints as B and C.

**Hint:** Ensure that you measure the length accurately using the ruler.

2. **Construct the Arcs to Find the Third Vertex:**
- Place the compass pointer at point B and set the compass width to 4.5 cm. Draw an arc above the line segment BC.
- Without changing the compass width, place the compass pointer at point C and draw another arc that intersects the first arc. Label the intersection point as A.

**Hint:** Make sure the compass remains at the same width while drawing arcs from both points.

3. **Draw the Triangle:**
- Connect points A, B, and C using a ruler to form triangle ABC.

**Hint:** Use a straight edge to ensure the lines are straight.

4. **Construct the Angle Bisectors:**
- To find the incenter of the triangle, draw the angle bisector of angle A. To do this, place the compass pointer at point A, and draw arcs that intersect both sides of the angle (AB and AC).
- Label the intersection points as D and E. Now, without changing the compass width, draw arcs from points D and E to intersect each other. Label the intersection point as F.
- Repeat this process for angles B and C to find points G and H.

**Hint:** Ensure that the arcs from points D and E intersect at a single point.

5. **Finding the Incenter:**
- The point where all three angle bisectors intersect is the incenter (point O) of triangle ABC.

**Hint:** Use a pencil to mark the intersection clearly.

6. **Draw the Inscribed Circle:**
- With the compass pointer at point O, measure the distance from O to any side of the triangle (for example, to side BC) to find the radius (let's call it r).
- Keeping the compass width set to r, draw a circle centered at point O.

**Hint:** Ensure the compass is steady while drawing the circle to maintain an even shape.

7. **Measure the Radius of the Circle:**
- Use a ruler to measure the radius of the inscribed circle from point O to the point where the circle touches side BC.

**Hint:** Make sure to measure straight from the center to the edge of the circle.

### Summary of Radius Measurement:
- The radius of the inscribed circle is approximately 1.3 cm or 1.4 cm, depending on the precision of your drawing.