Circle : Constructions - Class 10 MCQ

The side length of an equilateral triangle can be calculated from its circumradius (R) using the formula: Side length = \(R \times \sqrt{3}\). This relationship arises from the properties of equilateral triangles and their circumcircles, highlighting the proportionality between radius and side length.

When constructing a tangent to a circle from a point on its circumference, the angle formed between the radius at the point of contact and the tangent line is always 90°. This is a fundamental property of tangents and plays a crucial role in geometric constructions involving circles.

To construct the circumcircle of a regular hexagon, one can draw the perpendicular bisectors of any two adjacent sides. The intersection point of these bisectors serves as the center of the circumcircle, which will pass through all vertices of the hexagon. This method is efficient due to the regularity of the hexagon.

To construct an inscribed circle within a triangle, one must draw the angle bisectors of any two angles. The point where these bisectors intersect is known as the incentre, from which a circle can be drawn that touches all three sides of the triangle. This circle is known as the incircle.

The circumcentre of a triangle is defined as the point where the perpendicular bisectors of its sides intersect. This point is equidistant from all vertices of the triangle, and it serves as the center of the circumcircle, which passes through all three vertices.

The incentre of a triangle is found by constructing the angle bisectors of at least two of its angles. The intersection of these bisectors is the incentre, which is equidistant from all three sides of the triangle. This point serves as the center of the inscribed circle.

The lengths of the tangents drawn from an exterior point to a circle are always equal. This can be proven using the properties of right triangles formed by the radius and the tangent lines. This equality is a key property that simplifies many constructions in geometry.

The definition of a tangent states that a tangent line touches a circle at exactly one point. At this point of contact, the radius drawn to the point of contact is perpendicular to the tangent line, resulting in an angle of 90°. This fundamental property is crucial for constructing tangents correctly.

The interior angle of a regular hexagon can be calculated using the formula \((2n - 4) / n \times 90°\), where \(n\) is the number of sides. Substituting \(n = 6\) gives \((2 \times 6 - 4) / 6 \times 90° = 120°\). This consistent angle is one of the characteristics that make hexagons unique and efficient for tiling.

A tangent line to a circle is defined by its property of touching the circle at exactly one point, known as the point of tangency. This unique characteristic differentiates tangents from other lines that may intersect the circle at multiple points.