Placement Aptitude (Quant, Logical, Verbal) Module 16 – Non-verbal and spatial reasoning
Mirror images and shape transformations
Find mirror and water images of letters, words and clocks, and apply translation, rotation, scaling and reflection to shapes in the order given.
What you will learn
- Find mirror and water images of figures, letters and clocks
- Apply translation, rotation, scaling and reflection to shapes
- Find a figure embedded in a complex figure and assemble shapes from pieces
Before you start
What you’ll learn · Before you start
What you will learn
- Find mirror and water images of figures, letters and clocks
- Apply translation, rotation, scaling and reflection to shapes
- Find a figure embedded in a complex figure and assemble shapes from pieces
Before you start
On this page
Mirror images and water images are reflections. An upright mirror beside a figure swaps left and right and keeps top and bottom; a water image, the reflection in still water below the figure, swaps top and bottom and keeps left and right. Shape transformations are the moves that GATE’s spatial aptitude syllabus lists with mirroring: translation slides a shape, rotation turns it about a point, scaling enlarges or shrinks it, and reflection flips it over a line. Items ask for the image after one or more of these moves, the mirror image of a word or a clock, a shape hidden in a larger figure, or the pieces that make a target. Follow a few reference points, such as corners and the ends of strokes, through every move in the order given.
Mirror images, water images and letters
| Mirror image | Water image | |
|---|---|---|
| Where the mirror is | Upright, beside the figure | Level, below the figure |
| What swaps | Left and right | Top and bottom |
| A point (x, y) goes to | (−x, y) | (x, −y) |
| Capitals that look unchanged | A H I M O T U V W X Y | B C D E H I K O X |
| A word | Order reversed, each letter flipped | Order kept, each letter upside down |
In mirror images of words, each letter keeps its distance from the mirror on the other side, so the word reads backwards. Letters with a vertical line of symmetry survive the flip, which is why WOW looks the same in a mirror. The lists hold for a plain geometric font; in many other fonts the bowls of B are unequal, so fair items fix one font.
The mirror image of a clock
A clock and its mirror image
Text description of the diagram
The left clock has tick marks for the hours and a small triangle at 12. Its long minute hand points to the 8 and its short hour hand sits two-thirds of the way from the 3 to the 4: 3:40.
A dashed upright line between the clocks is the mirror. In the right clock, its mirror image, the minute hand points to the 4 and the hour hand sits one-third of the way from the 8 to the 9, where a real clock has it at 8:20.
A mirror on the line through 12 and 6 sends a hand pointing θ degrees clockwise from 12 to 360° − θ. Both hands move this way, so a time t becomes 12:00 − t, worked as 11:60 − t to avoid borrowing: 3:40 becomes 8:20, and 3:40 + 8:20 = 12:00 is the check. Only 12:00 and 6:00, on the mirror line, look unchanged. Read 12:10 as 0:10 before subtracting, which gives 11:50.
Shape transformations and their order
| Move | A point (x, y) goes to | What stays the same |
|---|---|---|
| Translation by (a, b) | (x + a, y + b) | Size, shape and facing |
| Quarter turn clockwise | (y, −x) | Size, shape and the order of corners |
| Half turn | (−x, −y) | Size, shape and the order of corners |
| Scaling by k | (kx, ky) | Shape and angles; areas scale by k² |
| Reflection in the y-axis | (−x, y) | Size and shape; the order of corners reverses |
Turn then mirror, or mirror then turn
Text description of the diagram
The top row starts from an upright letter F: a long stroke on the left with two arms pointing right, the top arm longer than the middle one. An arrow labelled "quarter turn" leads to the F turned a quarter turn clockwise: its long stroke now lies along the top, with two arms hanging down, the longer one at the right end. An arrow labelled "mirror" leads to the result: the same shape mirrored left to right, so the longer arm now hangs at the left end.
The bottom row starts from the same F. An arrow labelled "mirror" leads to a mirrored F, its arms pointing left from a long stroke on the right. An arrow labelled "quarter turn" leads to the result: a long stroke along the bottom with two arms standing up from it, the longer one at the right end. The two results differ: one has its long stroke at the top, the other at the bottom.
Coordinates, with x to the right and y upwards and the origin at the centre of the move, make every move exact. The order matters. A quarter turn clockwise followed by the mirror sends (x, y) to (y, −x) and then to (−y, −x), a reflection in the diagonal from top left to bottom right. The mirror followed by the turn gives (y, x), a reflection in the other diagonal, and the diagram shows the two different results for an F. So apply shape transformations exactly in the order the item gives.
Embedded figures and assembling
An embedded-figure item hides a simple shape inside a complex one. The shape is there only if every edge of it is drawn in the figure at the same size and slant, so trace it from one distinctive corner and count its edges. An assembling item asks which pieces make a target shape. Check the areas first: on squared paper the pieces’ squares must add up to the target’s, which rules out most wrong sets at once.
Five solved examples
- A word in a mirror. TOMATO reads OTAMOT, and every letter survives the flip: T, O, M and A all have a vertical line of symmetry.
- A clock. 9:15 in a mirror reads 11:60 − 9:15 = 2:45.
- A point. (4, 1) turned a quarter turn clockwise is (1, −4); mirrored after that, it is (−1, −4).
- Scaling. A 2 cm by 3 cm rectangle scaled by 2 becomes 4 cm by 6 cm: the perimeter doubles from 10 cm to 20 cm and the area grows four times, from 6 cm² to 24 cm².
- An area check. Pieces of 5, 6 and 4 squares cannot make a 4 × 4 square, because 5 + 6 + 4 = 15, not 16.
Checking by program
# Mirror images of a clock, worked by reflecting both hands, checked on every minute of the dial; then the F shape
# of the diagram turned and mirrored in both orders, with coordinates (x to the right, y upwards).
def hands(t):
"""Angles of the hour and minute hands, in degrees clockwise from 12, t minutes after 12:00."""
return (t / 2) % 360, (6 * t) % 360
def time_of(t):
h, m = divmod(t % 720, 60)
return f"{h or 12}:{m:02d}"
def minutes(text):
h, m = map(int, text.split(":"))
return (h % 12) * 60 + m
def mirror_reading(t):
"""The real time whose hands sit where the mirrored hands sit, or None if no real time does."""
hour, minute = hands(t)
mirrored = ((360 - hour) % 360, (360 - minute) % 360)
found = [u for u in range(720) if hands(u) == mirrored]
return found[0] if found else None
for text in ["3:40", "4:25", "12:10", "6:00", "9:15"]:
t = minutes(text)
shown = mirror_reading(t)
print(f"{text} in a mirror reads {time_of(shown)}; 11:60 minus {text} is {time_of(720 - t)}")
real = sum(1 for t in range(720) if mirror_reading(t) == (720 - t) % 720)
print(f"Every minute of the dial: {real} of 720 mirror images are real times, each equal to 12:00 minus the time")
# The F of the diagram: corners of its outline.
F = [(-10, -15), (-10, 15), (11, 15), (11, 9), (-4, 9), (-4, 3), (6, 3), (6, -3), (-4, -3), (-4, -15)]
def turn(p): # a quarter turn clockwise about the origin
x, y = p
return (y, -x)
def mirror(p): # an upright mirror along the y-axis
x, y = p
return (-x, y)
turn_then_mirror = [mirror(turn(p)) for p in F]
mirror_then_turn = [turn(mirror(p)) for p in F]
print(f"Turn, then mirror: {turn_then_mirror[:3]} ...")
print(f"Mirror, then turn: {mirror_then_turn[:3]} ...")
print(f"Same figure: {sorted(turn_then_mirror) == sorted(mirror_then_turn)}")
print(f"Turn, then mirror sends each (x, y) to (-y, -x): {all(q == (-p[1], -p[0]) for p, q in zip(F, turn_then_mirror))}")
print(f"Mirror, then turn sends each (x, y) to (y, x): {all(q == (p[1], p[0]) for p, q in zip(F, mirror_then_turn))}")Output
3:40 in a mirror reads 8:20; 11:60 minus 3:40 is 8:20 4:25 in a mirror reads 7:35; 11:60 minus 4:25 is 7:35 12:10 in a mirror reads 11:50; 11:60 minus 12:10 is 11:50 6:00 in a mirror reads 6:00; 11:60 minus 6:00 is 6:00 9:15 in a mirror reads 2:45; 11:60 minus 9:15 is 2:45 Every minute of the dial: 720 of 720 mirror images are real times, each equal to 12:00 minus the time Turn, then mirror: [(15, 10), (-15, 10), (-15, -11)] ... Mirror, then turn: [(-15, -10), (15, -10), (15, 11)] ... Same figure: False Turn, then mirror sends each (x, y) to (-y, -x): True Mirror, then turn sends each (x, y) to (y, x): True
Recorded with Python 3.14.8 on macOS 26 arm64. To run it yourself: mise exec python@3.14.8 -- python3 mirror_clock.py
Runs on this device, in your browser. The first run downloads Python (about 13.5 MB), which is kept for the next runs.
Your run, in this browser
The program reflects both hands for all 720 minutes of a 12-hour dial and finds a real time every time, always 12:00 minus the time shown. It also turns and mirrors the F in both orders and confirms that the results differ.
Reverse Text, Mirror & Upside Down Type a word and see its mirrored and upside-down versions side by side, to test your predictions.Your turn
The exercise reads a clock in a mirror; the ten questions after it are untimed.
Exercise · Easy · Python
Read a clock in a mirror
An analogue clock stands in front of an upright mirror. Write mirror_time(time), which takes the time the clock shows as text such as "3:40" (hours 1 to 12, minutes 00 to 59) and returns the time its mirror image shows, in the same form: the hour without a leading zero, the minutes as two digits.
The mirror swaps left and right, so each hand lands as far from 12 on the other side of the dial. Times on the line from 12 to 6 look the same in the mirror.
Starter code · clock.py
def mirror_time(time):
"""The time a mirror image of an analogue clock shows, as "h:mm"."""
# Replace this line with your code.
return timeThe sample tests · test_clock.py
from clock import mirror_time
def test_lesson_example():
"""The lesson's clock: 3:40 reads 8:20"""
assert mirror_time("3:40") == "8:20"
def test_quarter_past():
"""9:15 reads 2:45"""
assert mirror_time("9:15") == "2:45"
def test_just_after_twelve():
"""12:10 reads 11:50"""
assert mirror_time("12:10") == "11:50"
def test_on_the_twelve_six_line():
"""6:00 and 12:00 look the same"""
assert mirror_time("6:00") == "6:00"
assert mirror_time("12:00") == "12:00"
def test_minutes_keep_two_digits():
"""1:05 reads 10:55, and 11:59 reads 12:01"""
assert mirror_time("1:05") == "10:55"
assert mirror_time("11:59") == "12:01"A hint
Turn the time into minutes after 12:00 first: (hours % 12) * 60 + minutes, so that 12:10 becomes 10. The mirror image is 12 hours minus that, and % 720 keeps 12:00 at 12:00. Turn the result back with divmod(..., 60), and write an hour of 0 as 12.
Results of the sample tests
| Test | Result | Details |
|---|
What your code printed
The sample tests run on this device, in your browser (Pyodide): nothing is sent to mysmartcopilot.com. The first run downloads Python (about 13.5 MB), which is kept for the next runs. A check in your browser is feedback for you, not proof that the code is right for every input.
Ten questions, untimed
10 questions about this lesson. Every answer and why it is right is on the page, behind “Show the answer”. Your score stays in this browser.
Mirror, water and clock items with full working Pro: opens with a Pro, Premium or Ultimate pass
Word, display and clock items repeat a few patterns, and the hardest turn on a fact that shortcuts skip: a water image of a clock never shows a real time. These twelve original items work each pattern and settle that case with numbers.
In this part:
- Words and letters, worked
- Seven-segment displays, worked segment by segment
- Clocks in mirrors, worked
- Water images and upside-down clocks: two readings, with the numbers
- A move and a slide in both orders
1 runnable example (Python) · about 3 minutes
Opening this part with your pass…
This part opens with any MySmartCoPilot pass.
- Pro pass · 1 week₹79$2.99
- 30 days₹179$5.99
- 1 year₹1,699$45
One payment; a pass never renews. The price shown is the total you pay: no fees or charges are added at checkout.
Passes can’t be bought right now, so this part can’t be opened at the moment. The free part above stands on its own. If you have a pass, enter your licence key.
It also opens every other in-depth part in Learn and every Pro tool. Sales are final, except where the law gives you a right to withdraw, for example in the EU and the UK (refund policy).
Transformations in depth: order, corner order, embedded figures and pieces Pro: opens with a Pro, Premium or Ultimate pass
Options in transformation items differ in ways a quick look misses: a reflection where a turn was asked, a shape at the wrong slant, pieces of the right area that do not fit. This part gives the rules that separate them, each checked by a program.
In this part:
- Which moves give the same result in either order, with a proof
- The corner-order invariant: telling a turn from a reflection
- Embedded figures, checked edge by edge
- Assembling pieces: the area check, then the fit
2 diagrams · 1 runnable example (Python) · about 4 minutes
Opening this part with your pass…
This part opens with any MySmartCoPilot pass.
- Pro pass · 1 week₹79$2.99
- 30 days₹179$5.99
- 1 year₹1,699$45
One payment; a pass never renews. The price shown is the total you pay: no fees or charges are added at checkout.
Passes can’t be bought right now, so this part can’t be opened at the moment. The free part above stands on its own. If you have a pass, enter your licence key.
It also opens every other in-depth part in Learn and every Pro tool. Sales are final, except where the law gives you a right to withdraw, for example in the EU and the UK (refund policy).
A ten-minute set on mirrors and transformations, with a score rubric Pro: opens with a Pro, Premium or Ultimate pass
The set below has ten original items on mirrors, clocks and moves, at a minute apiece. Answer them in one ten-minute sitting before you read the key and the score bands.
In this part:
- Ten items for ten minutes
- Answers with the shortest working
- Scoring the set, with a rubric
1 runnable example (Python) · about 3 minutes
Opening this part with your pass…
This part opens with any MySmartCoPilot pass.
- Pro pass · 1 week₹79$2.99
- 30 days₹179$5.99
- 1 year₹1,699$45
One payment; a pass never renews. The price shown is the total you pay: no fees or charges are added at checkout.
Passes can’t be bought right now, so this part can’t be opened at the moment. The free part above stands on its own. If you have a pass, enter your licence key.
It also opens every other in-depth part in Learn and every Pro tool. Sales are final, except where the law gives you a right to withdraw, for example in the EU and the UK (refund policy).
Interview questions
-
Fresher · Warm-up
What is the difference between a mirror image and a water image of a figure, and what happens to a word written in capitals in each?
A mirror image comes from an upright mirror beside the figure: left and right swap while top and bottom stay, so a point (x, y) goes to (−x, y). A water image comes from a level surface below the figure: top and bottom swap while left and right stay, so (x, y) goes to (x, −y). In a mirror a word reads backwards and each letter is flipped left to right; letters with a vertical line of symmetry, such as A, M, O and T, look unchanged, so TOMATO reads OTAMOT. In water the letters keep their order and each is turned upside down; letters with a horizontal line of symmetry look unchanged, so BOX looks the same in a plain geometric font.
-
Fresher
A wall clock shows 4:25. What does its mirror image show, why does subtracting from 11:60 work, and what happens to the rule at 12:10 and at 6:00?
-
Fresher
A flag is turned a quarter turn clockwise and then reflected in an upright mirror. Is that the same as reflecting first and turning second? Show it with coordinates, and name the single move that each order equals.
-
Fresher
Why does the water image of an analogue clock never show a real time, and how should you answer an item that asks what a clock reads in water?
Key takeaways
- An upright mirror swaps left and right; a water image swaps top and bottom.
- A clock’s mirror image shows 12:00 minus the time, worked as 11:60 − t.
- A quarter turn and a mirror give different results in the two orders, so follow the order given.
- Turns keep the order of labelled corners and reflections reverse it.
References
- GATE 2027 General Aptitude syllabus (spatial aptitude: translation, rotation, scaling, mirroring, assembling) (Indian Institute of Technology Madras)
- Combined Graduate Level Examination 2026 notice, indicative Tier-I syllabus (embedded figures, space orientation) (Staff Selection Commission)
- Ganita Prakash, Grade 6, Chapter 9: Symmetry (lines of symmetry and reflection) (National Council of Educational Research and Training (NCERT))
- Built-in functions: divmod (Python Software Foundation)
- itertools: combinations (Python Software Foundation)
Related tools
Report a problem with this lesson
Kept only in this browser. Your Learn progress