幾何與角

In the figure, $A$, $B$ and $C$ are points lying on the circle. $AB$ is a diameter of the circle. $DB$ is the tangent to the circle at $B$. If $ACD$ is a straight line with $AC=4$ and $CD=2$, then $AB=$

If $\triangle ABC$ is an obtuse-angled triangle, which of the following points must lie outside $\triangle ABC$? I. The centroid of $\triangle ABC$ II. The circumcentre of $\triangle ABC$ III. The orthocentre of $\triangle ABC$

If the plane figure above is rotated anticlockwise about the point $O$ through $90^\circ$, which of the following is its image?

In the figure, $ABC$ and $AED$ are straight lines. If $AB=8\text{ cm}$, $BC=4\text{ cm}$ and $CD=9\text{ cm}$, then $BE=$

In the figure, $O$ is the centre of the circle $ABC$. If $\angle OBC=50^\circ$ and $\angle ACO=20^\circ$, then $\angle BOA=$

In the figure, $O$ is the centre of the circle. $A$ and $B$ are points lying on the circle. If $AOC$ is a straight line and $BC$ is a tangent to the circle, then the radius of the circle is

In the figure, $ABC$ is an acute-angled triangle, $AB=AC$ and $D$ is a point lying on $BC$ such that $AD$ is perpendicular to $BC$. Which of the following must be true? I. The circumcentre of $\triangle ABC$ lies on $AD$. II. The orthocentre of $\triangle ABC$ lies on $AD$. III. The centroid of $\triangle ABC$ lies on $AD$.

In the figure, $ABCD$ is a circle. $AB$ produced and $DC$ produced meet at $E$. If $AC$ and $BD$ intersect at $F$, then $\angle ABD=$

In the figure, $ABCD$ is a circle. If $AC$ is a diameter of the circle and $AB=BD$, then $\angle CAD=$

If $AC=BD$ and $AB\parallel DC$, how many pairs of similar triangles are there in the figure?