Triangle DEF Is a Scale Copy of Triangle ABC: What Is the Length of Side EF? The Ratio Method Behind the 12 cm Answer

The length of side EF is 12 centimeters, which is option b — but only when the scale factor from triangle ABC to triangle DEF is 3 and the corresponding side BC measures 4 cm. The question as written is missing that ratio, so check your diagram or problem statement for the scale factor or a pair of matching sides before locking in an answer.

This is the standard trap in scale-copy problems: students jump to one of the four answer choices without first confirming which sides correspond. A scale copy means every side of triangle DEF is the same multiple of the matching side in triangle ABC, and that single multiplier is what turns the missing measurement into arithmetic. Without it, all four listed lengths — 2.4, 12, 25, and 15 centimeters — are arithmetically reachable from different scale factors, which is exactly why the problem supplies one somewhere in the diagram, the table, or the preceding sentence.

What "Scale Copy" Means and Which Sides Actually Correspond

A scale copy (some curricula say "scaled copy" or "similar figure") is a figure whose side lengths are all multiplied by one common number, called the scale factor, while its angles stay identical. For triangles, that means angle A matches angle D, angle B matches angle E, and angle C matches angle F, so side AB corresponds to DE, side BC corresponds to EF, and side AC corresponds to DF. The letter order in the names is the correspondence key — DEF is named to line up with ABC position by position.

That ordering rule is the single most useful fact here. If the problem says triangle DEF is a scale copy of triangle ABC, then side EF occupies the third-and-second position, exactly like side BC. So EF is never compared to AB or AC. Students who match EF with AC get a plausible-looking number and a wrong answer, because AC and DF are the pair that correspond.

One important boundary condition: a scale copy can enlarge or shrink. A scale factor of 3 triples every side, a factor of 0.5 halves them, and a factor of 1 produces a congruent triangle that is technically still a scale copy. The scale factor is always stated as "from ABC to DEF" or "from DEF to ABC" — direction matters, and reversing it flips the multiplier to its reciprocal.

The Illustrative Mathematics curriculum, which popularized this exact phrasing, teaches scale copies in grade 7 alongside the constant of proportionality. The Common Core standard 7.G.A.1 requires students to solve problems involving scale drawings, and scale-copy triangle problems are the standard entry point. If your worksheet uses the phrase "scale copy," it almost certainly comes from that curriculum family, where the scale factor is always given as a whole number or simple fraction.

Why the Answer Is 12 cm When the Scale Factor Is 3

When the scale factor from ABC to DEF is 3 and side BC measures 4 cm, side EF measures 12 cm because 4 × 3 = 12. That's the whole computation — one multiplication. The other three answer choices exist because they represent the three common mistakes: dividing instead of multiplying (4 ÷ 3 gives roughly 1.33, and if a student starts from a different side they can land near 2.4), adding the scale factor (4 + 3 = 7, not an option), or using the wrong corresponding side entirely.

Here is the worked correspondence for that scenario:

Notice that 15 cm appears in that table as side DF — and 15 is answer choice d in your question. That's precisely how distractors are built: the test writer computes a real side of triangle DEF and offers it as if it were EF. If a student solves for DF by mistake (matching AC to EF instead of DF), they select 15 and feel confident.

The triangle above is the classic 3-4-5 right triangle, which is why these numbers show up so often in scale-copy exercises. Tripling it produces a 9-12-15 triangle, still a right triangle, still similar. That relationship — same shape, proportional sides, preserved angles — is the definition being tested.

Using the Scale Factor in Reverse to Find It

If your problem gives you side EF and side BC but hides the scale factor, divide: scale factor = EF ÷ BC. Suppose EF measures 12 cm and BC measures 4 cm, then 12 ÷ 4 = 3. Suppose instead EF is 2.4 cm and BC is 4 cm, the factor is 0.6, meaning DEF is a shrunken copy at 60 percent. The direction of the division is always "new side ÷ original side," and swapping the order gives you the reciprocal — 4 ÷ 12 = 1/3, the factor from DEF back to ABC.

This reverse method is the reliable check when an answer feels wrong. Take answer choice a, 2.4 cm. For EF to be 2.4 while BC is 4, the scale factor from ABC to DEF must be 0.6. That's a legitimate scale factor, so 2.4 cm isn't an impossible number — it's just the answer to a different version of the problem. The same logic applies to 25 cm, which needs a factor of 6.25, and to 15 cm, which needs 3.75. All three are mathematically valid for some triangle; none is valid for the specific triangle the question describes.

Here's a quick decision path for any scale-copy triangle problem:

1: Write the two triangle names side by side and draw arrows between matching letters: A→D, B→E, C→F. 2: Find the one pair of corresponding sides where both lengths are known. 3: Divide new length by original length to get the scale factor. 4: Multiply the known side that corresponds to the missing side by that factor. 5: Sanity-check: if the copy is larger, the missing side must be larger than its partner in ABC.

If step 2 fails because no pair is fully known, the problem intends you to use the scale factor stated in words or implied by a grid — check the instructions above the figure.

What 2.4, 25, and 15 Centimeters Each Represent

The distractor set in this question is unusually clean, which makes it a good teaching example. Each wrong answer maps to a specific student error, and naming those errors is faster than re-deriving the problem.

The 15 cm distractor deserves special attention because it is a real side of the triangle in the worked example above — it's the hypotenuse DF. When a student sets up the proportion EF/AC = 3 instead of EF/BC = 3, they get 5 × 3 = 15. The error is correspondence, not arithmetic, and correspondence errors are the single most common failure on scale-copy items in the Illustrative Mathematics grade 7 unit assessments. If you got 15, your multiplication was right and your side-matching was wrong.

The 2.4 cm option catches a different failure: treating the scale factor as a divisor. Some students see "scale copy" and assume the copy is smaller, so they compute 4 ÷ a number. If they divide 4 by something approximating 1.67, or divide 12 by 5, they drift toward 2.4. Choice c, 25 cm, is the least principled of the four — it rarely comes from a clean error and usually signals a misread diagram measurement.

Setting Up the Proportion Without Crossing Sides

The proportion form of this problem is EF/BC = DE/AB = DF/AC, all equal to the scale factor. Written with the numbers from the 3-4-5 example: EF/4 = 9/3. Cross-multiplying gives 3 × EF = 36, so EF = 12. The proportion method and the multiply-by-factor method give identical results; the proportion is safer when the scale factor isn't explicitly given, because it forces you to write the correspondence.

A common setup slip is inverting one ratio: writing EF/BC = AB/DE, which cross-multiplies to the wrong equation. Keep both ratios pointing the same direction — either both "DEF over ABC" or both "ABC over DEF." Mixing directions produces the reciprocal of the correct factor, and on a multiple-choice test that result usually isn't listed, which sends students back to guess.

For students who prefer a mental model, think of the scale factor as a coupon multiplier rather than a discount. A "3× coupon" triples every side. Three applications of that thinking — one per side — fill in the whole triangle. The angles never change, so angle E always equals angle B no matter what the factor is, and that fact is worth writing down alongside the side lengths because many scale-copy problems ask about angles in a second part.

Teachers using the Open Up Resources 7th-grade materials will recognize this as Unit 1, Lesson 2 territory, where students sort cards into scaled copies and non-copies. The non-copies are typically figures where two sides scaled by 3 but a third scaled by 4 — a deliberately broken figure that tests whether students check all pairs, not just one.

Checking Your Answer Against the Picture and the Choices

Two fast checks catch most wrong answers before you bubble anything in. First, compare sizes: if triangle DEF is visibly larger than ABC in the diagram, EF must be bigger than BC — that immediately eliminates 2.4 cm. Second, test the factor against a second side: if the problem gave you AB and DE, confirm that DE ÷ AB equals the factor you used for EF.

Diagrams in these problems are usually not to scale, and the instructions often say so. Don't measure with a ruler. Use the labeled numbers only, and treat the drawing as a rough orientation aid — which side looks longest, which angle looks right. If the labeled numbers and the drawing disagree about which side is longest, trust the numbers; the drawing is decorative.

If your answer isn't among the choices, the error is almost always correspondence. Rework the problem by writing the letter-matching arrows before doing any arithmetic. On the four options given here, a student who lands on 9 cm (side DE in the 3-4-5 example) has solved a real side but not the requested one — a sign the question was read too quickly.

When the Scale Factor Is a Fraction or Decimal

Fractional scale factors produce the shrinking answers, and they trip up students who expect whole numbers. If the scale factor from ABC to DEF is 1/2 and BC is 4 cm, then EF is 2 cm. If it's 3/5, EF is 2.4 cm — which is exactly where answer choice a comes from in a shrinking version of this problem. That's worth knowing because it means 2.4 is not a random distractor; it's the correct answer to a variant where DEF is smaller than ABC.

This connects to a broader proportionality idea: the ratio of any pair of corresponding sides equals the ratio of the perimeters. If the scale factor is 3, the perimeter of DEF is three times the perimeter of ABC. For the 3-4-5 triangle, ABC's perimeter is 12 cm and DEF's is 36 cm. Area behaves differently — it scales by the square of the factor, so DEF's area is nine times ABC's. That asymmetry between linear and area scaling is a frequent follow-up question and a frequent source of error.

Keep units consistent throughout. If BC is given in centimeters and the scale factor is unitless (it always is, since it's a ratio), then EF comes out in centimeters automatically. Mixing millimeters into a centimeters problem without converting is a silent error that produces answers like 120 instead of 12 — not among these choices, but common on longer tests.

How This Question Appears on Tests and What the Graders Want

On the Smarter Balanced and PARCC-style assessments that use scale-copy language, the item typically appears in a two-part format: first identify the scale factor, then compute the missing side. Partial credit is often available for the factor alone, so writing "scale factor = 3" before the final number is a scoring-safe habit. The answer itself is usually worth one point and the justification another.

Constructed-response versions ask students to explain why the triangles are similar, and the expected reasoning names two things: all three pairs of corresponding sides share the same ratio, and all three pairs of corresponding angles are equal. Stating only the side ratio is usually accepted for scale copies but marked incomplete for similarity proofs in grade 8, where the angle criterion becomes mandatory.

A practical exam tip tied to this exact question: when four numeric choices are offered and two of them are related by a clean factor (12 and 15 differ by a factor of 1.25; 12 and 2.4 differ by 5), the test writer built the distractors around the correct value. That pattern isn't proof, but it's a useful tiebreaker when time is short and the correspondence is ambiguous in a poorly printed figure.

For homeschool parents grading this at the kitchen table, the fastest check is to ask the student which side in DEF matches BC. If they point to EF without hesitating and can say "third letter to second letter," the reasoning is sound regardless of arithmetic slips. Fix the arithmetic separately from the correspondence — they're different skills and they fail independently.

FAQ

Is the answer 12 cm or 15 cm if the scale factor is 3?

It's 12 cm for side EF, because EF corresponds to BC (4 cm) and 4 × 3 = 12. The 15 cm figure is side DF, which corresponds to AC (5 cm) and equals 5 × 3. Both numbers are real sides of triangle DEF in the standard 3-4-5 example, so the question is really testing whether you matched the correct pair of sides rather than whether you can multiply.

What if no scale factor is given anywhere in the problem?

Then the problem is incomplete as stated, and all four choices become defensible under different factors — 2.4 needs 0.6, 12 needs 3, 15 needs 3.75, and 25 needs 6.25. Check the diagram labels, any table above the question, and the sentence before it; Illustrative Mathematics items virtually always supply the factor or a full pair of corresponding sides. If genuinely absent, ask your teacher which pair of sides was intended to correspond.

Do the angles change when a triangle is scale-copied?

No. A scale copy preserves all three angle measures exactly, which is why the triangles are similar rather than merely proportional. Angle D equals angle A, angle E equals angle B, and angle F equals angle C regardless of whether the factor is 0.5, 3, or 6.25. Only the side lengths and the perimeter change, and area changes by the square of the factor.

How is a scale copy different from a congruent triangle?

A congruent triangle is a scale copy with a factor of exactly 1, so every side and angle matches. A scale copy with any other factor — larger or smaller — is similar but not congruent. On tests, "scale copy" usually signals a non-1 factor, and the answer choices will reflect a changed size, so expect to multiply or divide rather than copy numbers straight across.

What's the fastest way to eliminate wrong choices on this type of question?

Compare the missing side's partner in the original triangle to the answer choices: if DEF is drawn larger, EF must exceed BC, which rules out 2.4 cm immediately. Then test the remaining candidates by dividing each by BC to see which produces a clean, consistent factor across a second known pair of sides. A candidate that requires a factor matching no other side pair is a distractor.

Does the order of letters in "triangle DEF" really control which sides correspond?

Yes, and it's the most reliable rule available. D matches A, E matches B, and F matches C by position, so DE pairs with AB, EF pairs with BC, and DF pairs with AC. Test writers follow this convention strictly, and violating it in your setup is the most common cause of a wrong answer on scale-copy items — especially the 15 cm trap.

If the scale factor is 3, what is the ratio of the areas?

The area ratio is 9 to 1, the square of the linear factor, so triangle DEF's area is nine times triangle ABC's. For the 3-4-5 example, ABC's area is 6 square centimeters (half of 3 × 4) and DEF's is 54 square centimeters. Linear dimensions scale by the factor; areas scale by its square; volumes, in three dimensions, scale by its cube.