CDS Mathematics Plane Geometry: Triangles and Circles Questions

ABCD is a quadrilateral with sides $AB=9$ cm, $BC=40$ cm, $CD=28$ cm and $DA=15$ cm and one of the diagonals $AC=41$ cm. Which of the following statements is/are correct?
I. The vertices $A,B,C$ and $D$ of the quadrilateral lie on a circle.
II. The area of triangle $ACD$ is $126{\text{cm}}^2$.
Select the answer using the code given below:

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. $\sqrt{3}AB=(2+\sqrt{3})PQ$

Select the answer using the code given below:

$ABCD$ is a quadrilateral such that $AC$ bisects $\angle A$ and $\angle C$. Is $\frac{AB}{DC}$ equal to $\frac{AD}{BC}$ ?

Statement-I : AB is parallel t o D C
Statement-ll : AD is parallel to BC

In a triangle $ABC$, $\angle A=30^{\circ }$, $AB=7\mathrm{cm}$ and $AC=12\mathrm{cm}$. What is the area of the triangle $ABC$?

ABC is a triangle right angled at B. Further, (AB+BC) exceeds AC by 10 units. If the perimeter of the triangle is 60 units, then what is the area of the triangle?

The difference between the two acute angles in a right-angled triangle is $\pi /12$ radian. One of the acute angles of the triangle is

ABC is a triangle right angled at B. P is the midpoint of AB and Q is the midpoint of BC. Consider the following : (1) AQ = $\sqrt{73}$ units (2) CP = $\sqrt{52}$ units Which of the above is/are required to determine the area of the triangle?

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?

ABC is an isosceles triangle with

AB = AC = 10 units. If the area of the triangle is 48 square units, then what is the length of the base BC?

Statement I:

The length of BC is an even integer.

Statement II:

The height of the triangle is greater than the length of half of the base.

Which one of the following is correct in respect of the above Question and the Statements?

Let X, Y and Z be the midpoints of the sides BC, CA and AB of a triangle ABC respectively. Consider the following statements :

I. The quadrilateral AZXY is a parallelogram.

II. The area of the quadrilateral AZXY is half of the area of the triangle ABC.

Which of the statements given above is/are correct?

Consider the following angles :

I. 4°

II. 5°

III. 6°

IV. 8°

How many of the above can be the exterior angle of a regular polygon?

ABCD is a quadrilateral in which $AD=BC$. What is $\angle ABC+\angle ADC$ equal to? Statement-I: $AB$ is parallel to $DC$. Statement-II: $AD$ is not parallel to $BC$. Which one of the following is correct in respect of the above Question and the Statements?

$ABC$ is a triangle right angled at $B$. $D$ is a point on $AC$ such that $BD$ is perpendicular to $AC$. If $AB=p$ and $BC=\sqrt{3}p$, then what is $BD$ equal to?

The sides of a triangle are $11\mathrm{cm}$, $60\mathrm{cm}$ and $61\mathrm{cm}$. What is the area of the triangle formed by joining the mid-points of the sides of the triangle?

A pendulum swings through an angle of $30^{\circ }$ and its end describes an arc of length $55\mathrm{cm}$. What is the length of the pendulum? (Take $\pi =22/7$)

A triangle PQR is inscribed in a circle with its centre at O. A tangent PT is drawn at P such that $\angle QPT=36^{\circ }$. What is $\angle POQ$ equal to?

Let AD be the altitude of a triangle ABC. If (AB + AC) = p, (AB - AC) = q and (BD - CD) = r, then what is BC equal to?

P, Q, R and S are respectively the mid-points of sides $AB$, $BC$, $CD$, $DA$ of a rhombus $ABCD$. These points are joined to form a quadrilateral $PQRS$. Is the quadrilateral cyclic ?

Statement-I : $\angle ABC$ = 120°

Statement-II: $\angle BCD$ = 60°

ABCD is a cyclic quadrilateral. AB and DC are produced to meet at P. Which of the following statements is/are correct? I. $△PAD$ is similar to $△PBC$ II. $\angle PAD+\angle PDA=\angle PBC+\angle PCB$ Select the answer using the code given below:

In a circle of radius 14 cm, APB is a shorter arc and P is the midpoint of the arc. Let C be the midpoint of the chord AB and PC = 7 cm. What is the length of the chord AP?