Cyclic Quadrilateral MCQ Quiz - Objective Question with Answer for Cyclic Quadrilateral - Download Free PDF
Last updated on Jun 12, 2025
Latest Cyclic Quadrilateral MCQ Objective Questions
Cyclic Quadrilateral Question 1:
ABCD is a cyclic quadrilateral such that ∠B = 112°. The tangents at A and C meet at a point P. What is the measure of ∠APC?
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 1 Detailed Solution
Given:
The ∠B = 112°
Formula used:
Opposite angles of the cyclic quadrilateral = 180°
Calculation:
In given cyclic quadrilateral ABCD
⇒ ∠ABC + ∠ADC = 180°
⇒ ∠ADC = 180° - 112° = 68°
Since PA is tangent on the circle at point A and AC is the cord which is subtending the angle ∠D = 68°
The angle between a tangent and a chord of a circle is equal to the angle subtended by the chord in the alternate segment of the circle.
In this case, the tangent at point A (line segment PA) and chord AC subtend ∠PAC in the alternate segment, which is equal to the angle ∠D subtended by the same chord AC. Therefore, ∠PAC = ∠D = 68°.
⇒ ∠PAC = ∠D = 68°
Also,
⇒ ∠PAC = ∠PCA, (As PA and PC are the tangents of A and C)
⇒ ∠PAC = ∠PCA = ∠ADC = 68°
In ΔPAC
⇒ ∠PAC + ∠PCA + ∠APC = 180°
⇒ 68° + 68° + ∠APC = 180°
⇒ ∠APC = 180° - 136° = 44°
∴ The required result will be 44°.
Cyclic Quadrilateral Question 2:
In a quadrilateral ABCD, angles A, B, C and D are respectively (3x - 10°), (x + 30°), (2x + 30°) and (2x - 10°). Then, this quadrilateral is a
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 2 Detailed Solution
Given:
The angles of quadrilateral ABCD are:
- Angle A = 3x-10°
- Angle B = x+30°
- Angle C = 2x+30°
- Angle D = 2x-10°
The sum of the interior angles of any quadrilateral is \( 360^\circ \).
- Set up the equation for the sum of angles:
3x-10 + x+30 + 2x+30 + 2x-10 = 360° - Simplify the equation: 8x + 40 = 360°
- Solve for
x " id="MathJax-Element-363-Frame" role="presentation" style="position: relative;" tabindex="0"> :
8x = 320°
x = 40° - Find each angle:
- Angle A = 110°
- Angle B = 70°
- Angle C = 110°
- Angle D = 70°
- Adjacent Angles:
- Angle A (110°) + Angle B (70°) = 180°
- Angle B (70°) + Angle C (110°) = 180°
- Angle C (110°) + Angle D (70°) = 180°
- Angle D (70°) + Angle A (110°) = 180°
The quadrilateral has:
- Two distinct pairs of adjacent sides where one pair is equal
- One pair of equal opposite angles (110° and 110°)
- The other pair of equal opposite angles (70° and 70°)
- Adjacent angles are supplementary (add up to 180°)
These properties are characteristic of a Parallelogram.
NOTE:
As per the official answer key, Option 3 and 4 both are correct.
Cyclic Quadrilateral Question 3:
The angles of cyclic quadrilaterals ABCD are: A = (6x + 10), B = (5x)°, C = (x + y)° and D =(3y - 10)°. The value of x and y is:
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 3 Detailed Solution
Formula used:
In a cyclic quadrilateral, opposite angles sum up to 180º.
A + C = 180º and B + D = 180º
Calculations:
⇒ (6x + 10) + (x + y) = 180
⇒ 7x + y = 170 .........(1)
⇒ (5x) + (3y - 10) = 180
⇒ 5x + 3y = 190 .........(2)
Now, solving the equations 1 and 2:
7x + y = 170
5x + 3y = 190
____________
21x + 3y = 510 ......(3)
5x + 3y = 190 .......(4)
Subtracting (4) from (3):
16x = 320 ⇒ x = 20º
Substitute x in (4):
5 x 20 + 3y = 190
⇒ 100 + 3y = 190
⇒ y = 30º
⇒ x = 20º and y = 30º
∴ The correct answer is option (2).
Cyclic Quadrilateral Question 4:
A quadrilateral PQRS is inscribed in a circle of centre O, such that PQ is a diameter and ∠ PSR = 120°. Find the value of ∠QPR.
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 4 Detailed Solution
Given:
PQRS is a cyclic quadrilateral, and ∠PSR = 120°
Calculation:
As we know, the sum of the either pair of the opposite angles of a cyclic quadrilateral is 180°
⇒ ∠PSR + ∠PQR = 180°
⇒ ∠PQR = 180° - 120° = 60°
In ΔPQR,
if PQ is diameter, then ∠PRQ = 90°
⇒ ∠PQR + ∠PRQ + ∠QPR = 180°
⇒ 60° + 90° + ∠QPR = 180°
⇒ ∠QPR = 180° – 150° = 30°
The correct answer is option 1.
Cyclic Quadrilateral Question 5:
In a cyclic quadrilateral the sum of opposite angles is:
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 5 Detailed Solution
Explanation:
In a cyclic quadrilateral, the sum of opposite angles is supplementary (i.e., they
add up to 180o).
Hence option 3 is correct.
Top Cyclic Quadrilateral MCQ Objective Questions
A circle touches all four sides of a quadrilateral PQRS. If PQ = 11 cm. QR = 12 cm and PS = 8 cm. then what is the length of RS ?
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 6 Detailed Solution
Download Solution PDFGiven :
A circle touches all four sides of a quadrilateral PQRS. If PQ = 11 cm. QR = 12 cm and PS = 8 cm
Calculations :
If a circle touches all four sides of quadrilateral PQRS then,
PQ + RS = SP + RQ
So,
⇒ 11 + RS = 8 + 12
⇒ RS = 20 - 11
⇒ RS = 9
∴ The correct choice is option 3.
PQRS is a cyclic trapezium where PQ is parallel to SR and PQ is the diameter. If ∠QPR = 40° then the ∠PSR is equal to:
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 7 Detailed Solution
Download Solution PDFGiven:
PQRS is a cyclic trapezium where PQ is parallel to RS.
PQ is the diameter & ∠QPR = 40°
Concept:
Angle made in a semicircle is a right angle.
The sum of the opposite angles of a cyclic trapezium is 180°.
Calculation:
In triangle PQR,
∠RPQ + ∠RQP + ∠QRP = 180° [Angle sum property]
⇒ 40° + ∠RQP + 90° = 180°
⇒ ∠RQP = 180° - 130° = 50°
∠RQP + ∠PSR = 180° [Supplementary Angles]
∴ ∠PSR = 180° - 50° = 130°
ABCD is a cyclic quadrilateral. Diagonals BD and AC intersect each other at E. If ∠BEC = 138° and ∠ECD = 35°, then what is the measure of ∠BAC?
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 8 Detailed Solution
Download Solution PDFGiven:
∠BEC = 138° and ∠ECD = 35°
Concept used:
In cyclic quadrilateral angles on the same arc are always same
Calculation:
∠BEC and ∠CED are on the same straight lines
∠BEC =138°
∠CED = 180° – 138°
⇒ ∠CED = 42°
In ΔCDE, ∠CED = 42° and ∠DCE = 35°
∠CDE = 180° - (42° + 35°)
∠CDE = 103°
∠BAC and ∠BDC are on the same arc BC
We know that In cyclic quadrilateral angles on the same arc are always same.
∠BAC = 103°
∴ The measure of ∠BAC is 103°
ABCD is a cyclic quadrilateral such that ∠B = 104°. The tangents at A and C meet at a point P. What is the measure of ∠APC?
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 9 Detailed Solution
Download Solution PDFGiven:
The ∠B = 104°
Formula used:
Opposite angles of the cyclic quadrilateral = 180°
Calculation:
In given cyclic quadrilateral ABCD
⇒ ∠ABC + ∠ADC = 180°
⇒ ∠ADC = 180° - 104° = 76°
Since PA is tangent on the circle at point A and AC is the cord which is subtending the angle ∠D = 76°
The angle between a tangent and a chord of a circle is equal to the angle subtended by the chord in the alternate segment of the circle.
In this case, the tangent at point A (line segment PA) and chord AC subtend ∠PAC in the alternate segment, which is equal to the angle ∠D subtended by the same chord AC. Therefore, ∠PAC = ∠D = 76°.
⇒ ∠PAC = ∠D = 76°
Also,
⇒ ∠PAC = ∠PCA, (As PA and PC are the tangents of A and C)
⇒ ∠PAC = ∠PCA = ∠ADC = 76°
In ΔPAC
⇒ ∠PAC + ∠PCA + ∠APC = 180°
⇒ 76° + 76° + ∠APC = 180°
⇒ ∠APC = 180° - 152° = 28°
∴ The required result will be 28°.
ABCD is a cyclic quadrilateral in which AB = 16 cm, CD = 18 cm and AD = 12 cm, and AC bisects BD. What is the value of AC.BD?
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 10 Detailed Solution
Download Solution PDFGiven:
AB = 16 cm
CD = 18 cm
AD = 12 cm
Concept used:
If diagonal PR bisects diagonal QS then
PQ × QR = PS × RS
In cyclic quadrilateral PQRS
PR × SQ = PQ × RS + PS × QR
Calculation:
According to the concept,
AB × BC = CD × AD
⇒ 16BC = 18 × 12
⇒ 16BC = 216
⇒ BC = 13.5 cm
Now,
Again according to the concept,
AC.DB = AB × CD + AD × BC
⇒ AC.DB = 16 × 18 + 12 × 13.5
⇒ AC.DB = 288 + 162
⇒ AC.DB = 450
∴ The value of AC.BD is 450.
In the given figure, PQ is a chord passing through the centre 'O' of the circle. Calculate ∠PQS.
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 11 Detailed Solution
Download Solution PDFConcept used:
The sum of the opposite angles of the cyclic quadrilateral is 180°.
∠PSQ = 90° because the angle is in a semi-circle.
Calculation:
∠SPQ = 180° - 110° = 70° (The sum of the opposite angles of the cyclic quadrilateral is 180°)
Now, In triangle PQS.
∠PSQ = 90° (Angle in semi-circle)
∠PQS = 180° - ( ∠PSQ +∠QPS)
⇒ 180° - (90° + 70°) = 20°
∴ The correct option is 3
PQRS is a cyclic quadrilateral in which PQ = x cm, QR = 16.8 cm, RS = 14 cm, PS = 25.2 cm, and PR bisects QS. What is the value of x?
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 12 Detailed Solution
Download Solution PDF
From the following figure
∠SQP = ∠OQP = ∠SRO
OQ = OS (Given)
∠QPR = ∠QSR = ∠OSR
∴ We can say that,
ΔPOQ∼ΔSOR
PQ/SR = OQ/OR
OQ/OR = x/14
Given, OQ = OS
OS/OR = x/14 --- (1)
As we know,
ΔPOS∼ΔQOR
OS/OR = PS/QR
OS/OR = 25.2/16.8 --- (2)
From equation (1) and equation (2)
x/14 = 25.2/16.8
x = (25.2 × 14)/16.8
∴ x = 21 cm
ABCD is cyclic quadrilateral. Sides AB and DC, when produced, meet at E, and sides BC and AD, when produced, meet at F. If ∠BFA = 60° and ∠AED = 30°, then the measure of ∠ABC is:
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 13 Detailed Solution
Download Solution PDFGiven:
ABCD is cyclic quadrilateral.
∠BFA = 60° and ∠AED = 30°
Calculation:
∠BFA = 60 & ∠BED = 30
Let ∠ABC = θ
∵ ABCD is a cyclic quadrilateral
∴ ∠ADC + ∠ABC = 180
∠ADC = 180 – θ ---- (1)
Now,
∠ADC + ∠FDC = 180 [straight line]
From Equation (1);
180 – θ + ∠FDC = 180
∠FDC = θ
In ΔDFC
∠FDC + ∠DFC + FCD = 180
θ + 60 + ∠FCD = 180
∠FCD = 180 – 60 – θ = 120 – θ
∠FCD = ∠BCE = (120 – θ) [Vertical opposite angle]
∠ABC + ∠CBE = 180 [straight line]
∠CBE = 180 – θ
In ΔBEC
∠CBE + ∠BCE + ∠BEC = 180
180 – θ + 120 – θ + 30 = 180
2θ = 330 – 180 = 150
θ = 150/2 = 75
Shortcut Trick
In triangle ABF
α + θ + 60 = 180
⇒ α + θ = 120 → (1)
In triangle ADE
α + π - θ + 30 = 180
⇒ α - θ + 30 = 0 → (2)
From Eqn (1) and (2), we have
α = 45° and θ = 75°
∴ ∠ABC = 75°
The length of two parallel sides of a trapezium are 53 cm and 68 cm respectively, and the distance between the parallel sides is 16 cm. Find the area of the trapezium.
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 14 Detailed Solution
Download Solution PDFArea of the Trapezium = 1/2 × (Sum of the parallel sides) × (Distance between parallel sides)
⇒ 1/2 × (53 + 68) × 16
⇒ 1/2 × 121 × 16
∴ Area of the Trapezium = 968 cm2
Sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E and sides AD and BC are produced to meet at F. If ∠ADC = 78° and ∠BEC = 52°, then the measure of ∠AFB is:
Answer (Detailed Solution Below)
Cyclic Quadrilateral Question 15 Detailed Solution
Download Solution PDFGiven:
∠ADC = 78° and ∠BEC = 52°
Concept:
The sum of a pair of opposite angles is 180° of a cyclic quadrilateral.
Calculation:
Now, In quadrilateral ABCD,
∠ADC + ∠ABC = 180° (Sum of opposite angles are 180°)
⇒ ∠ABC = 180° - 78° = 102°
∠EBC = 180° - 102° = 78° [Linear Pair]
In ΔBEC,
∠BEC + ∠EBC + ∠BCE = 180°
⇒ 52° + 78° + ∠BCE = 180°
⇒ ∠BCE = 180° - 130° = 50°
⇒ ∠BCE = ∠DCF = 50° (Vertically opposite angle)
∠ADE + ∠FDC = 180° [Linear Pair]
⇒ ∠FDC = 180° - ∠ADE = 180° - 78° = 102°
In ΔDCF,
∠FDC + ∠DCF + ∠DFC = 180°
102 + 50 + ∠DFC = 180°
∠DFC = 180° - 152° = 28°
∠DFC = ∠AFB = 28°
∴ The measure of ∠AFB is 28°.