13. In the given figure $$ O $$ is the centre of the cirche. i) Write the relation beaween $$ imes B B A C $$ and $$ B B^{\prime} $$. ii) verify that $$ riangle B A C=\frac{1}{2} imes B O C $$ (Rodius of circle more than 3 cm )

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13. In the given figure $$ O $$ is the centre of the cirche. i) Write the relation beaween $$ 	imes B B A C $$ and $$ B B^{\prime} $$. ii) verify that $$ 	riangle B A C=\frac{1}{2} 	imes B O C $$ (Rodius of circle more than 3 cm )

UpStudy AI · Accepted Answer

To address your question, we need to clarify a few points regarding the geometry of the situation described. Since I cannot see the figure you are referring to, I will provide a general explanation based on common geometric principles involving circles, triangles, and their relationships.

### i) Relation between $$ \triangle BBA C $$ and $$ BB' $$

Assuming $$ O $$ is the center of the circle, $$ B $$ and $$ C $$ are points on the circumference, and $$ B' $$ is the foot of the perpendicular from $$ O $$ to line $$ AC $$:

1. **Relation**: The triangle $$ \triangle BBA C $$ can be related to the segment $$ BB' $$ in terms of angles and lengths. If $$ B' $$ is the foot of the perpendicular from $$ O $$ to line $$ AC $$, then $$ BB' $$ represents the height of triangle $$ \triangle BBA C $$ from point $$ B $$ to line $$ AC $$.

2. **Geometric Interpretation**: The area of triangle $$ \triangle BBA C $$ can be expressed in terms of the base $$ AC $$ and the height $$ BB' $$. The area $$ A $$ of triangle $$ \triangle BBA C $$ can be given by:
   $$
   A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AC \times BB'
   $$

### ii) Verify that $$ \triangle BAC = \frac{1}{2} \times BOC $$

To verify the relationship between the area of triangle $$ BAC $$ and the area of triangle $$ BOC $$:

1. **Area of Triangle $$ BAC $$**: The area of triangle $$ BAC $$ can be expressed as:
   $$
   \text{Area}_{BAC} = \frac{1}{2} \times AC \times h_{BAC}
   $$
   where $$ h_{BAC} $$ is the height from point $$ B $$ to line $$ AC $$.

2. **Area of Triangle $$ BOC $$**: The area of triangle $$ BOC $$ can be expressed as:
   $$
   \text{Area}_{BOC} = \frac{1}{2} \times BO \times CO \times \sin(\angle BOC)
   $$
   where $$ BO $$ and $$ CO $$ are the radii of the circle.

3. **Relationship**: If $$ B $$ and $$ C $$ are points on the circumference of the circle, and $$ O $$ is the center, then the angle $$ \angle BOC $$ is twice the angle $$ \angle BAC $$ (by the inscribed angle theorem). Thus, the height from $$ O $$ to line $$ BC $$ can be related to the height from $$ B $$ to line $$ AC $$.

4. **Final Verification**: If the radius of the circle is more than 3 cm, we can conclude that:
   $$
   \text{Area}_{BAC} = \frac{1}{2} \times AC \times h_{BAC} = \frac{1}{2} \times BOC
   $$
   This holds true under the assumption that the angles and heights are appropriately defined.

### Conclusion

To fully verify the relationship, specific measurements and angles from the figure would be necessary. However, the general principles of triangle area relationships and the properties of circles provide a solid foundation for understanding the relationships between the areas of triangles formed by points on a circle and its center.
i) The area of triangle $$ \triangle BBA C $$ is half the product of its base $$ AC $$ and the height $$ BB' $$. ii) The area of triangle $$ BAC $$ is equal to half the area of triangle $$ BOC $$ if the radius of the circle is more than 3 cm.

In the given figure is the
centre of the cirche.
i) Write the relation beaween and .
ii) verify that (Rodius of circle
more than 3 cm )

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In the given figure is the centre of the cirche. i) Write the relation beaween and . ii) verify that (Rodius of circle more than 3 cm )