Geometry — CDS Elementary Mathematics PYQs

$ABC$ is a triangle right-angled at $C$. A semicircle is drawn on $AB$ as diameter. Let $P$ be any point on $AC$ produced such that $AP=AB=10$ cm. Further, $B$ and $P$ are joined. If $BC=8$ cm, then what is $BP$ equal to?

In a cyclic quadrilateral $ABCD$, the diagonal $AC$ bisects the angle $C$. Which of the following statements is/are correct?
I. $\angle ABD=\angle ADB$
II. The diagonal $BD$ is parallel to the tangent of the circle at $A$

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$ABC$ is a triangle in which $AB=AC$ and $D$ is any point on $BC$. Which of the following statements is/are correct?
I. $A{B}^2-A{D}^2=CD\times BD$
II. $B{C}^2+B{D}^2-D{C}^2=2BD\times BC$

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$ABC$ is a triangle right-angled at $B$ with sides $BC=a$, $CA=b$ and $AB=c$. Let $p$ be the length of the perpendicular from $B$ to $AC$. Which of the following statements is/are correct?
I. $p=\frac{ac}{\sqrt{a^2+c^2}}$
II. $pb=ac$

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$ABC$ is an equilateral triangle. Let $PQRS$ be a square inscribed in it such that $P$ is on $AB$ and $Q$ is on $AC$. Which of the following is/are correct?
I. $AP:PB=4:3$
II. $\sqrt{3}AB=(2+\sqrt{3})PQ$

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Let $O$ be the centre of a circle. Let $A$, $B$ and $C$ lie on the circle such that $\angle ABC=153^{\circ }$. What is the value of $\angle AOC$?

In a quadrilateral $ABCD$, the diagonals intersect at $O$. Let the area of the triangle $ABD$ be $p$. If $AO:OC=m:n$, then what is the area of the triangle $BCD$?

Let $ABCD$ be a parallelogram and $E$ be the midpoint of $BC$. The diagonal $BD$ and line segment $AE$ intersects at $F$. If $BF=2.4$ cm, then what is $BD$ equal to?

$ABC$ is a triangle right-angled at $C$. Let $P$ be the midpoint of $BC$. If $AP=4\sqrt{13}$ cm and $AB=20$ cm, then what is the perimeter of the triangle $ABC$?

$ABC$ is a triangle right-angled at $B$. Points $M$ and $N$ trisect $BC$. If $AM=6$ cm and $AN=9$ cm, then what is $AC$ equal to?