CDS Mathematics Plane Geometry: Triangles and Circles Medium 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:

$ABCD$ 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

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?

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. 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?

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 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?

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 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?

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?

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?

In a triangle ABC, $\angle A=\angle B-\angle C$. Is angle A acute?

Statement 1: ABC is not an obtuse-angled triangle.

Statement 2: $\angle C$ is acute.

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

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

(b) The Question can be answered by using either Statement alone

(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

(d) The Question can be answered even without using any of 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?

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°

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?

A Question is given followed by two Statements I and II. Consider the Question and the Statements.

Question :

In a triangle $ABC$ right angled at $B$, $AC=20$ cm. What is the circumradius of the triangle?

Statement-I :

$AB=12$ cm

Statement-II :

$BC=16$ cm

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

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

Two triangles $ABC$ right-angled at $A$ and $DBC$ right-angled at $D$ are drawn such that $AC$ and $DB$ intersect at $P$. If $AP=x$, $PC=y$ and $BP=z$, then what is $(AC+BD)$ equal to ?

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?