Right Triangles and Circle Geometry Questionbank

Which of the following statements correctly describes the relationship between a chord length $c$ and the diameter $d$ of the same circle?

A tangent is drawn to a circle at point $A$. A chord $AB$ is drawn such that it is parallel to another chord $BC$ (where $C$ is on the circumference). If $\angle ABC = 55^\circ$, find the size of $\angle BAC$.

Which of the following geometric descriptions correctly defines a sector of a circle?

In a right triangle, if $\tan \theta = \frac{3}{4}$, what is the value of $\sin \theta$?

A cyclic quadrilateral ${ABCD}$ is inscribed in a circle. If the side ${AD}$ is extended to a point ${E}$ outside the circle, which of the following statements is always true regarding the exterior angle ${\angle CDE}$?

If the central angle of a sector is $72^{\circ}$, what fraction of the total area of the circle is represented by this sector?

Points $A$, $B$, and $C$ lie on a circle with center $O$. If the angle at the circumference is $\angle ABC = 38^\circ$, what is the size of the central angle $\angle AOC$ standing on the same arc $AC$?

Sector A has a radius of $5 \text{ units}$ and a central angle of $2 \text{ radians}$. Sector B has a radius of $10 \text{ units}$ and a central angle of $1 \text{ radian}$. Which of the following statements is true regarding their areas?

In a right-angled triangle $ABC$ with the right angle at $C$, if the focus is switched from angle $\hat{A}$ to angle $\hat{B}$, which of the following statements is true?

Two tangents from an external point $T$ touch a circle of radius $r$ at points $A$ and $B$. If the distance from the centre $O$ to point $T$ is $2r$, what is the size of $\angle ATB$?