TACHS: Classification: find the mirror line
Learn to find the mirror line the three examples share, then check every dot in a choice for a partner directly across that line.
How do you pick the figure that shares the examples' symmetry when every choice has the same number of dots?
Short answer
Find the one mirror line that works for all three examples. Then, in each choice, find every dot's partner directly across that line. One missing partner rules a choice out.
Know first: Left, right, top and bottom, Rows and columns in a grid
Step 1 of 5
Symmetry means every dot has a partner
In a figure classification question, three example figures share a property, and you choose the one of five answer figures that shares it too. In symmetry questions, the examples are dot patterns with a mirror line. Fold a figure along that line and every dot lands exactly on another dot.
The line can be vertical (left half matches right half) or horizontal (top half matches bottom half). The choices usually have the same number of dots, so you have to check partners instead of counting.
| Mirror line | (row, column) pairs with | Example |
|---|---|---|
| Vertical, between columns 2 and 3 | (row, 5 − column) | (2, 1) pairs with (2, 4) |
| Horizontal, between rows 2 and 3 | (5 − row, column) | (1, 3) pairs with (4, 3) |
| A half turn (not a mirror) | (5 − row, 5 − column) | (1, 1) matches (4, 4) |
Describe each figure as dots in squares (row, column). The half-turn row is there because it fools people: it looks balanced but is not a mirror.
Step 2 of 5
Test the line, then test every dot
Worked example: Top-bottom symmetry
Example 1: dots at (1, 1), (2, 3), (3, 3), (4, 1).
Example 2: (1, 2), (2, 1), (2, 4), (3, 1), (3, 4), (4, 2).
Example 3: (1, 3), (1, 4), (2, 2), (3, 2), (4, 3), (4, 4).
Each choice below has six dots. Which belongs with the examples?
- A(1, 1), (1, 4), (2, 2), (2, 3), (4, 2), (4, 3)
- B(1, 1), (1, 2), (2, 4), (3, 1), (4, 3), (4, 4)
- C(1, 1), (1, 4), (2, 2), (3, 2), (4, 1), (4, 4)
- D(1, 1), (1, 4), (2, 2), (3, 3), (4, 1), (4, 4)
- E(1, 1), (1, 4), (2, 3), (3, 3), (3, 4), (4, 1)
- 1
Test a vertical line
In Example 1, (1, 1) would need a partner at (1, 4). There is none, so the line is not vertical.
- 2
Test a horizontal line
Rows 1 and 4 pair up, and rows 2 and 3 pair up. In all three examples every dot has its partner. The property is top-bottom symmetry.
- 3
Check the choices
(1, 1), (1, 4), (2, 2), (2, 3), (4, 2), (4, 3) mirrors left to right, but (1, 1) has no partner at (4, 1). The half-turn figure fails at (1, 1) too. In the near-misses, (3, 3) needs (2, 3) and (1, 4) needs (4, 4), and neither is there.
- 4
Confirm the answer
(1, 1) and (4, 1), (2, 2) and (3, 2), (1, 4) and (4, 4). Three full pairs.
Answer
Dots at (1, 1), (1, 4), (2, 2), (3, 2), (4, 1), (4, 4).
Find the wrong step
The examples have left-right symmetry. A student checks a choice with dots at (1, 1), (1, 4), (2, 2), (3, 3), (4, 1), (4, 4).
- 1
Pairs (1, 1) with (1, 4).
- 2
Pairs (4, 1) with (4, 4).
- 3
Pairs (2, 2) with (3, 3), because they look balanced around the center.
What went wrong
Those two match only after a half turn. Across a vertical line, (2, 2) needs a partner at (2, 3), in the same row.
The fix
A mirror partner is always in the same row for a vertical line. (2, 2) has no partner, so the choice is out.
Step 3 of 5
Check partners on your own
Check yourself · Question 1
A figure has dots at (1, 2), (1, 3), (2, 1), (2, 4), (3, 1), (4, 2), (4, 3). The mirror line is vertical, between columns 2 and 3. Which dot has no partner?
Answer: C
(3, 1) needs a partner at (3, 4), and there is no dot there. The others pair up: (1, 2) with (1, 3), (2, 1) with (2, 4), (4, 2) with (4, 3).
Check yourself · Question 2
Example 1: dots at (1, 2), (1, 3), (3, 1), (3, 4). Example 2: (1, 1), (1, 4), (2, 2), (2, 3), (4, 1), (4, 4). Example 3: (1, 1), (1, 4), (2, 1), (2, 4), (3, 2), (3, 3), (4, 2), (4, 3). Which choice belongs with the examples?
Answer: D
Every example mirrors left to right across the line between columns 2 and 3. In the answer, (1, 2) pairs with (1, 3), (2, 1) with (2, 4), and (4, 2) with (4, 3).
Trap answer: (1, 1), (1, 2), (2, 3), (3, 3), (4, 1), (4, 2)
Right property, wrong line
- Why it looks right
- It really is symmetric, with three neat pairs, so it feels like a match.
- Why it's wrong
- Its pairs are top and bottom. Every example pairs left and right, so the property is symmetry across a vertical line.
- The right answer
- (1, 2), (1, 3), (2, 1), (2, 4), (4, 2), (4, 3).
Common mistake
I checked a few dots, they had partners, and I picked that choice.
- Why it's tempting
- Near-miss choices are built so that most dots pair up. The figure looks balanced at a glance.
- Do this instead
- Check every dot. One unpaired dot is enough to rule a choice out.
Step 4 of 5
What to remember
Remember
- Find the one mirror line, vertical or horizontal, that works for all three examples.
- In each choice, every dot needs a partner directly across that line at the same distance.
- A figure that is symmetric across the other line, or only after a half turn, does not belong.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, which mirror line did the examples share, and which dot ruled out the closest wrong choice?