Geometry: Aptitude Questions with Answers
Geometry is a vast topic. Review Geometry Basic Concepts before jumping into the Questions:
Q. 1 If a line divides two sides of a triangle proportionally, then it is:
(A) a median
(B) an altitude
(C) a perpendicular bisector
(D) parallel to the third side
Check Solution
Ans: (D) parallel to the third side
By the Basic Proportionality Theorem (BPT), if a line divides two sides of a triangle proportionally, it is parallel to the third side.
Q. 2 You’re drawing a triangle where two of its angles are each 60°. What type of triangle have you created?
(A) Isosceles
(B) Right-angled
(C) Equilateral
(D) Scalene
Check Solution
Ans: (C) Equilateral
The sum of angles in a triangle is 180°. Since two angles are 60°, the third angle will also be 60°. Thus, all angles are equal, making the triangle equilateral.
Q. 3 Ramanujan is building a ramp. The sides of the ramp measure 6 cm and 8 cm. Find out the length of the diagonal support beam (the hypotenuse) to complete the design?
(A) 10 cm
(B) 12 cm
(C) 14 cm
(D) 9 cm
Check Solution
Ans: (A) 10 cm
By Pythagoras’ theorem:
$\text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2$
$\text{Hypotenuse}^2 = 6^2 + 8^2 = 36 + 64 = 100$
$\text{Hypotenuse} = \sqrt{100} = 10 \, \text{cm}$
Q. 4 The ratio of the areas of two similar triangles is:
(A) Equal to the ratio of their sides
(B) Equal to the square of the ratio of their sides
(C) Equal to the cube of the ratio of their sides
(D) None of these
Check Solution
Ans: (B) Equal to the square of the ratio of their sides
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Q. 5 In a circle, a chord is a line segment:
(A) Joining two points on the circle
(B) Passing through the center of the circle
(C) Parallel to the radius
(D) None of these
Check Solution
Ans: (A) Joining two points on the circle
A chord is defined as a line segment joining two points on the circumference of a circle.
Q. 6 The measure of each exterior angle of a regular polygon with 10 sides is:
(A) 30°
(B) 36°
(C) 45°
(D) 60°
Check Solution
Ans: B) 21 cm
The measure of each exterior angle of a regular polygon is:
$\text{Exterior Angle} = \frac{360^\circ}{\text{Number of sides}} = \frac{360^\circ}{10} = 36^\circ$
Q. 7 “The Racing Magic” takes 120 seconds to complete one lap around the track, while “The Charging Bull” can complete 40 laps in 60 minutes. If both start at the same point simultaneously, how many minutes will pass before they both arrive at the starting point together again for the second time?
(A) 3
(B) 6
(C) 9
(D) 12
Check Solution
Ans: D
Q. 8 In a right-angled triangle, if you were to draw the circumcircle (the circle passing through all three vertices), how does the radius of that circle relate to the hypotenuse?
(A) Half of the hypotenuse
(B) Hypotenuse
(C) Twice the hypotenuse
(D) None of these
Check Solution
Ans: (A) Half of the hypotenuse
The circumradius R of a right triangle is equal to half the length of its hypotenuse.
Q. 9 Two tangents drawn to a circle from an external point are:
(A) Perpendicular
(B) Parallel
(C) Equal in length
(D) Unequal in length
Check Solution
Ans: (C) Equal in length
Tangents drawn from an external point to a circle are always equal in length.
Q. 10 In ΔABC, if AB² + BC² = AC². Identify the type of triangle this equation represents:
(A) Scalene
(B) Isosceles
(C) Right-angled
(D) Equilateral
Check Solution
Ans: (C) Right-angled
If the square of one side equals the sum of the squares of the other two sides, the triangle is right-angled (by Pythagoras’ theorem).
Q. 11 A , B , C and D are points on a line , with D the midpoint of segment BC . The lengths of segments AB , AC and BC are 10 , 2 , 12 , respectively . What is the length of segment AD ?
(A) 2
(B) 4
(C) 6
(D) 10
Check Solution
Ans: B
Q. 12 A school library wall is 25 meters in length and needs to fit a combination of desks and bookshelves, with at least one of each. Each desk measures 3 meters and each bookshelf measures 2 meters. What is the length of the unused space along the wall if the greatest number of desks and bookshelves are used?
(A) 1
(B) 2
(C) 3
(D) 0
Check Solution
Ans: D
Let’s first keep one desk and one bookshelf as one of each is mandatory. Then we are left with length = 25 – 3 – 2 = 20
Now since bookshelf measures smaller compared to desk, to maximize the number we must keep only bookshelves in remaining area. We can keep 10 bookshelves completely occupying this 20m space, leaving 0 space.
Q. 13 Points Q and R are on segment PS , which has length 24 . If PQ is half as long as QS , and QR is 3 times as long as RS , how long is RS ?
(A) 2
(B) 3
(C) 4
(D) 8
Check Solution
Ans: C
Let PQ = x
QS = 2x
So, x + 2x = 24
-> x = 8
PQ = 8
QS = 16
P…………..Q…………………………..S
Let RS = y
QR = 3y
so y + 3y = 16
-> y = 4
RS = 4
QR = 12
P…….8…..Q………….12……….R…..4….S
Q. 14 In the track scenario described above, 8 competitors are racing to complete the course in the least amount of time. There is a 1.2-meter gap between each lane, from Runner 1 to Runner 2 and so on. If each athlete must stick to their own lane, with Runner 1 in the first lane and Runner 8 in the eighth lane, how far ahead should Runner 8 start from Runner 1 to ensure both run an equal distance to the finish line?
(A) 37
(B) 51
(C) 53
(D) 64
Check Solution
Ans: C
Refer Mensuration Questions: https://www.learntheta.com/aptitude-questions-mensuration/