CAT Previous Year Questions | Polygon
- CAT 2022 QA Slot 1 | Geometry - Quadrilaterals & Polygons CAT Question
A trapezium ABCD has side AD parallel to BC. ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of this trapezium is 36 cm, then its area, in sq cm, is
EXPLANATION
- CAT 2022 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
Regular polygons A and B have number of sides in the ratio 1 : 2 and interior angles in the ratio 3 : 4. Then the number of sides of B equals
EXPLANATION
Correct Answer: 10
Let the number of sides of polygons A and B be n and 2n, respectively
The number of sides of polygon B is 2*5, i.e. 10
- CAT 2022 QA Slot 3 | Geometry - Quadrilaterals & Polygons CAT Question
The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number o possible lengths of the fourth side is
(a) 5
(b) 4
(c) 3
(d) 6
EXPLANATION
The correct answer is A:5
To determine the possible lengths of the fourth side of the quadrilateral,we need to consider the triangle inequality theorem.According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.This principle also holds for quadrilaterals.
Given the lengths of the three sides as 1cm, 2cm, and 4cm, we can consider all the possible combinations of these sides as two sides of the quadrilateral.Let's check each case:
1cm+2cm> 4cm (True)
1cm+4cm> 2cm (True)
2cm+4cm> 1 cm (True)
Since all three combinations satisfy the triangle inequality theorem,any of them can form a valid triangle,and consequently,a quadrilateral with the given side lengths.
Now,let's consider the possible lengths of the fourth side for each combination:
For 1cm+2cm> 4cm:
The fourth side can have a length between |1cm-2cm|+1 and 1cm+2cm-1.
This gives us a range of possible lengths: 1 to 2cm.
For 1cm+4cm> 2cm:
The fourth side can have a length between |1cm - 4cm|+1 and 1cm+ 4cm- 1.
This gives us a range of possible lengths: 2 to 4cm.
For 2cm+ 4cm> 1cm:
The fourth side can have a length between |2cm- 4cm|+ 1 and 2cm+ 4cm- 1.
This gives us a range of possible lengths: 3 to 5cm.
Now,let's combine the ranges from all three cases: 1-2cm,2-4cm, and 3-5cm.
The possible integer lengths for the fourth side are: 1, 2, 3, 4, 5.
Therefore, the total number of possible lengths for the fourth side is 5.
- CAT 2021 QA Slot 1 | Geometry - Quadrilaterals & Polygons CAT Question
Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. If T is the mid point of CD, then the length of AT, in cm, is
(a) √14
(b) √12
(c) √13
(d) √15
EXPLANATION
T is the midpoint of CD in the given regular hexagon ABCDEF
CT=DT=2/2=1
Draw AT, AD and AC
By the property of hexagon,
AD=2×BC=2×2=4=2×=2×2=4
∠ABC=∠BCD=120∘∠=∠=120∘
We have, AB=BC=CD=2
∠ACD=120∘−30∘=90∘∠=120∘−30∘=90∘
ACD is a right triangle and AT is its median
AC2=AD2−CD2= 42 - 22 = 12
Using Apollonius's theorem,
AC2+AD2=2(AT2+DT2)
12 + 42 = 2(AT2+12)
AT = √13
Ans: ≈3.605551 cm
- CAT 2021 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
The sides AB and CD of a trapezium ABCD are parallel, with AB being the smaller side. P is the midpoint of CD and ABPD is a parallelogram. If th difference between the areas of the parallelogram ABPD and the triangle BPC is 10 sq cm, then the area, in sq cm, of the trapezium ABCD is
(a) 20
(b) 30
(c) 40
(d) 25
EXPLANATION
It is given that:
DP =x
∴ AB =x
Now DP=CP So CD = 2x
Now, if we denote the height of the trapezium as 'h,'
we can express the area of parallelogram ABPD as 'xh'
Now, based on the given condition
- CAT 2021 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is
(a) √13 + √12
(b) √37 + √13
(c) (√13 + √12)/2
(d) (√37 + √13)/2
EXPLANATION
Answer: A
All the sides of the rhombus are equal.
The area of a rhombus is 12 cm2.
Considering d1 to be the length of the longer diagonal, d2 to be the length of the shorter diagonal.
The area of a rhombus is
d1*d2 = 24.
- CAT 2021 QA Slot 3 | Geometry - Quadrilaterals & Polygons CAT Question
A park is shaped like a rhombus and has area 96 sq m. If 40 m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires alon its two diagonals, at the rate of ₹125 per m, is
EXPLANATION
- CAT 2021 QA Slot 3 | Geometry - Quadrilaterals & Polygons CAT Question
Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. If the angle ∠ADC is equal to 30° the the area of the parallelogram is sq. cm. is
(a) 25(√3 + √15)/2
(b) 25(√5 + √15)
(c) 25(√3 + √15)
(d) 25(√5 + √15)/2
EXPLANATION
- CAT 2020 QA Slot 1 | Geometry - Quadrilaterals & Polygons CAT Question
A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is
(a) 5π/18
(b) 2π/15
(c) 3π/25
(d) 6π/25
EXPLANATION
- CAT 2020 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle and the area, R, of the rectangle, both in s cm, satisfy the relationship R = T². If the sides of the rectangle are in the ratio 1 : 3, then the length, in cm, of the longer side of the rectangle, is
(a) 27
(b) 24
(c) 18
(d) 21
EXPLANATION
Let the sides of the rectangle be "a" and "3a" m.
Hence the perimeter of the rectangle is 8a. Let the side of the equilateral triangle be "m" cm.
Hence the perimeter of the equilateral triangle is "3m" cm. Now we know that 8a+3m=90......(1)
Hence a=9 and the longer side of the rectangle will be 3a=27cm
- CAT 2020 QA Slot 3 | Geometry - Quadrilaterals & Polygons CAT Question
In a trapezium ABCD, AB is parallel to DC, BC is perpendicular to DC and ∠BAD = 45°. If DC = 5 cm, BC = 4 cm, the area of the trapezium in sq.cm is
EXPLANATION
Given, BC = DE = 4
CD = BE = 5
12.CAT 2019 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is 3/2 times each interior angle of A, the each interior angle, in degrees, of a regular polygon with a + b sides is
EXPLANATION
- CAT 2018 QA Slot 1 | Geometry - Quadrilaterals & Polygons CAT Question
Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD an CG is longer than EB, then the ratio of length of EB to that of CG is
(a) 2 : 5
(b) 4 : 9
(c) 3 : 8
(d) 1 : 3
EXPLANATION
It is given that EFGH is also a square whose area is 62.5% of that of ABCD. Let us assume that E divides AB in x : 1. Because of symmetry we can se that points F, G and H divide BC, CD and DA in x : 1. Because of symmetry we can se that points F, G and H divide BC, CD and DA in x : 1.
Let us assume that 'x+1' is the length of side of square ABCD.
- CAT 2018 QA Slot 1 | Geometry - Quadrilaterals & Polygons CAT Question
In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P be a point on CD such that AP i perpendicular to CD. Then the area, in sq cm, of triangle APD is
(a) 12√3
(b) 24√3
(c) 18√3
(d) 32√3
EXPLANATION
Area of the parallelogram ABCD = (base) (height)
= CD × AP = 72 sq.cm.
⇒ 72 = 9 × (AP) ⇒ AP = 8cm
- CAT 2018 QA Slot 1 | Geometry - Quadrilaterals & Polygons CAT Question
Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth o ABCD?
(a) 24, 12
(b) 24, 10
(c) 25, 10
(d) 25, 9
EXPLANATION
We know that AC is the diameter and ABC = 90°. AC = 2*13 = 26 cm.
In right angle triangle ABC,
Let us check with the options. Therefore, we can say that option A is the correct answer.
Therefore, we can say that option A is the correct answer.
- CAT 2018 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
A parallelogram ABCD has area 48 sqcm. If the length of CD is 8 cm and that of AD is s cm, then which one of the following is necessarily true?
(a) s ≠ 6
(b) s ≥ 6
(c) 5 ≤ s ≤ 7
(d) s ≤ 6
EXPLANATION
Answer:
As,sin0 < 1
Step-by-step explanation:
Area of parellogram ABCD= 2*Area of triangle ACD
48= 2* area of triangle ACD
Area of triangle of ACD =24 because 48/2=24
(1/2) * CD * DA * SinADC =24
=(1/2) *8* DA *SinADC =24
Ad *SinADC =6
ANSWER
As,sin0 < 1
- CAT 2018 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
The area of a rectangle and the square of its perimeter are in the ratio 1 ∶ 25. Then the lengths of the shorter and longer sides of the rectangle are i the ratio ?
(a) 1 : 3
(b) 2 : 9
(c) 1 : 4
(d) 3 : 8
EXPLANATION
Let 'a' and 'b' be the length of sides of the rectangle, (a > b)
Area of the rectangle = a*b
Perimeter of the rectangle = 2*(a+b)
=>a*b/(2*(a + b))2 = 1/25
=> 25ab = 4(a + b)2
=> 4a2 - 17ab + 4b2 = 0
=> (4a - b)(a - 4b) = 0
=> a = 4b or b/4
We initially assumed that a > b, therefore a ≠ 4. Hence, a = 4b => b: a = 1: 4
- CAT 2017 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is
(a) 3√2
(b) 3
(c) 4
(d) √3
EXPLANATION
AC is base of an isoceles triangle ABC where AB and BC are side lengths of a regular hexagon. Interior angles of hexagon are 120∘ each, therefore base angles are 30∘ each.
By cosine rule,
- CAT 2017 QA Slot 2 | Geometry - Quadrilaterals & Polygons CAT Question
ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
EXPLANATION
Following the conditions of question, we get the figure as:
∠COD = 1200 (Given)
Then, ∠DAC = 600 (Half of ∠COD i.e. 1200)
Therefore, ∠BAD = 300+600=900
Hence, ∠BCD = 1800 - ∠BAD (Opposite angles of a cyclic quadrilateral are supplementary angles).
∠BCD = 1800–900=900
Question 1 of 31
Consider a square ABCD with midpoints E, F, G, H of AB, BC, CD and DA respectively. Let L denote the line passing through F and H. Consider points and Q, on L and inside ABCD, such that the angles APD and BQC both equal 120°. What is the ratio of the area of ABQCDP to the remaining area insid ABCD?
Each question is followed by two statements A and B. Answer each question using the following instructions. Mark (1) if the question can be answered by using statement A alone but not by using statement B alone. Mark (2) if the question can be answered by using statement B alone but not by using statement A alone. Mark (3) if the question can be answered by using both the statements together but not by using either of the statements alone. Mark (4) if the question cannot be answered on the basis of the two statements. Rahim plans to draw a square JKLM with a point O on the side JK but is not successful. Why is Rahim unable to draw the square? A. The length of OM is twice that of OL. B. The length of OM is 4 cm.
An equilateral triangle BPC is drawn inside a square ABCD. What is the value of the angle APD in degrees?
Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90° o concave if the internal angle is 270°. If the number of convex corners in such a polygon is 25, the number of concave corners must be
In the figure below, ABCDEF is a regular hexagon and ∠AOF = 90°. FO is parallel to ED. What is the ratio of the area of the triangle AOF to that of th hexagon ABCDEF?
A square tin sheet of side 12 inches is converted into a box with open top in the following steps – the sheet is placed horizontally. Then, equal size squares, each of side x inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of box. If x is an integer, then what value of x maximizes the volume of the box?
Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?
The length of the circumference of a circle equals the perimeter of a triangle of equal sides, and also the perimeter of a square. The areas covered by the circle, triangle, and square are c, t, and s, respectively. Then,
Let S1 be a square of side a. Another square S2 is formed by joining the mid-points of the sides of S1. The same process is applied to S2 to form yet another square S3, and so on. If A1, A2, A3… are the areas and P1, P2, P3… are the perimeters of S1, S2, S3… respectively, then the ratio P1+P2+P3+…..
In the following figure, the area of the isosceles right triangle ABE is 7 sq.cm. If EC = 3BE, then the area of rectangle ABCD (in sq. cm.) is
In the above diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the area of the triangle CEF and that of the rectangle?
Based on the figure below, what is the value of x, if y = 10?
ABCD is a rhombus with the diagonals AC and BD intersecting at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?
The figure below shows two concentric circles with centre O. PQRS is a square inscribed in the outer circle. It also circumscribes the inner circle touching it at points B, C, D, and A. What is the ratio of the perimeter of the outer circle to that of polygon ABCD?
There is a circle of radius 1 cm. Each member of a sequence of regular polygons S1(n), n = 4, 5, 6, …, where n is the number of sides of the polygon, is circumscribing the circle, and each member of the sequence of regular polygons S2(n), n = 4, 5, 6, …, where n is the number of sides of the polygon, is inscribed in the circle. Let L1(n) and L2(n) denote the perimeters of the corresponding polygons of S1(n) and S2(n), then {L1(13)+2π} is
A rectangle PRSU is divided into two smaller rectangles PQTU and QRST by the line TQ. PQ = 10 cm, QR = 5 cm, and RS = 10 cm. Points A, B, F are within rectangle PQTU, and points C, D, E are within the rectangle QRST. The closest pair of points among the pairs (A, C), (A, D), (A, E), (F, C), (F, D), (F, E), (B, C), (B, D), (B, E) are 10√3 cm apart.
AB > AF > BF ; CD > DE > CE ; and BF = 6√5 cm. Which is the closest pair of points among all the six given points?
Directions: Each question is followed by two statements I and II. Mark: 1. if the question can be answered by any one of the statements alone but cannot be answered by using the other statement alone. 2. if the question can be answered by using either statement alone. 3. if the question can be answered by using both the statements together but cannot be answered by using either statement alone. 4. if the question cannot be answered even by using both the statements together. Mr. X starts walking northwards along the boundary of a field from point A on the boundary, and after walking for 150 m reaches B, and then walks westwards, again along the boundary, for another 100 m when he reaches C. What is the maximum distance between any pair of points on the boundary of the field? I. The field is rectangular in shape. II. The field is a polygon, with C as one of its vertices and A as the mid-point of a side.
Four identical coins are placed in a square. For each coin, the ratio of area to circumference is the same as the ratio of circumference to area. Then find the area of the square that is not covered by the coins.
The adjoining figure shows a set of concentric squares. If the diagonal of the innermost square is 2 units, and if the distance between the corresponding corners of any two successive squares is 1 unit, find the difference between the areas of the eighth and the seventh squares, counting from the innermost square.
In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter to the longer side?
In the given figure, EADF is a rectangle, and ABC is a triangle whose vertices lie on the sides of EADF, and AE = 22, BE = 6, CF = 16, and BF = 2. Find the length of the line joining the mid-points of the sides AB and BC.
If ABCD is a square and BCE is an equilateral triangle, what is the measure of ∠DEC?
The figure shows the rectangle ABCD with a semicircle and a circle inscribed inside it as shown. What is the ratio of the area of the circle to that of the semicircle?
Which of the following values of x do not satisfy the inequality (x² – 3x + 2 > 0) at all?
PQRS is a square. SR is a tangent (at point S) to the circle with center O, and TR = OS. Then the ratio of the area of the circle to the area of the square is
In the adjoining figure, AC + AB = 5AD, and AC – AD = 8. Then the area of the rectangle ABCD is
Four friends start from four towns, which are at the four corners of an imaginary rectangle. They meet at a point that falls inside the rectangle after traveling distances of 40, 50, and 60 meters. The maximum distance that the fourth could have traveled is (approximately)…
Data is provided followed by two statements – I and II – both resulting in a value, say I and II. As your answer, Type 1, if I > II. Type 2, if I < II. Type 3, if I = II. Type 4, if nothing can be said. In ΔACD, AD = AC, and ∠C = 2∠E. The distance between parallel lines AB and CD is h. Then I. Area of parallelogram ABCD II. Area of ΔADE
Consider the five points comprising the vertices of a square and the intersection point of its diagonals. How many triangles can be formed using these points?
Let the consecutive vertices of a square S be A, B, C & D. Let E, F & G be the mid-points of the sides AB, BC & AD, respectively, of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest to