Similar hands you equal angles and a fixed ratio; congruent hands you everything.
What is the value of the area?
Triangle ABC is similar to triangle DEF, where A corresponds to D, B corresponds
to E, and C corresponds to F. The length of AB is 12 and
the length of DE is 18. If the area of triangle ABC is
40 square inches, what is the area, in square inches, of
triangle DEF?
The sides scale by k = 1812 = 32, so the areas scale by k² = 94, giving 40 × 94 = 90. A applies the side ratio straight to the area (40 × 32), the single most common slip in this topic. C multiplies by 3, reading “32” as “three times”. D scales twice — 40 × 32 = 60, then 60 × 94 = 135.
1
Similar fixes the shape; congruent fixes the shape and the size.
Know which one the stem handed you — they give you different facts.
similar ~
angles equal, sides in one ratio
a photo and its enlargement
congruent ≅
angles equal, sides equal
the ratio is just 1
AA is the criterion the SAT actually uses: two pairs of equal angles make the
triangles similar. The third pair is then forced, since all three total 180°.
a shared angletwo right anglesvertical anglesparallel lines
Those four are where the equal angles come from, almost every time. For congruence
the list is SSS, SAS, ASA, AAS and HL — AAA is not on it, because equal angles fix
the shape but say nothing about the size.
2
The correspondence sentence tells you which part pairs with which.
“Triangle ABC is similar to triangle DEF, where A corresponds to D” — read it straight across, first letter with first letter.
In △ABC
pairs with
In △DEF
∠A = 55°
↔
∠D = 55°
∠C = 40°
↔
∠F = 40°
AC = 8
↔
DF = 12
AB
↔
DE
Every side pair scales by the same k = 128 = 1.5. The angles do not scale at all —
they are equal, full stop. AC pairs with DF because A↔D and C↔F; pair AC with DE
instead and every number after that is wrong.
“Note: Figure not drawn to scale”
When that line sits under a figure, the drawing is only a label-holder: a side
marked 9 may be drawn longer than one marked 40. Trust the sentence, never the
picture. What a figure still states in words — a right-angle square, a tick mark,
a point lying on a segment — is given and is safe to use.
3
Congruent triangles copy every part, even when one is flipped.
Matching tick marks mean equal lengths; the orientation is a distraction.
△ABC ≅ △DEF
A↔D, B↔E, C↔F
AB = DE, BC = EF, AC = DF
matching ticks
∠A = ∠D, ∠B = ∠E, ∠C = ∠F
the angles come free too
A reflection changes nothing you can measure. When the stem says congruent and names
the correspondence, read the length you need straight off the other triangle — the
scale factor is 1, so there is no work to do.
4
For a missing side, write the proportion before you touch the numbers.
Triangle ABC is similar to triangle DEF, where A corresponds to D, B to E and C to F. AB = 16, DE = 24 and BC = 22. What is the length of EF?
ABDE =
BCEF →
1624 =
22EF
16 × EF = 24 × 22
cross-multiply
16 × EF = 528
24 × 22 = 528
EF = 33
528 ÷ 16
EF = 33
Or find the scale factor once: k = 2416 = 1.5, then EF = 22 × 1.5 = 33. The
calculator is allowed the whole way through, so a factor like 1.5 costs nothing.
Keep each triangle on its own level
Small over big on the left means small over big on the right. Writing
1624 = EF22 inverts one side and gives EF = 14.67 — shorter than the 22 it is
supposed to be an enlargement of.
5
A line parallel to one side cuts off a similar triangle.
In triangle ABC, D lies on AB, E lies on AC, and DE is parallel to BC. AD = 12, DB = 8 and DE = 18. What is the length of BC?
△ADE ~ △ABC
shared ∠A, and ∠ADE = ∠ABC from the parallels
AB = 12 + 8 = 20
the whole side, not just the piece
k = 2012 = 53
big over small
BC = 18 × 53 = 30
DE scaled up
AD pairs with AB, never with DB
The small triangle is ADE and the big one is ABC, so the ratio is AD/AB = 1220.
Using AD/DB = 128 instead gives BC = 12 — a base shorter than the DE sitting
inside it. Sanity-check the direction every time.
6
Lengths scale by k, areas by k².
any length
× k
perimeter
× k
area
× k²
k = 1812 = 32
DE over AB, from the opening problem
k² = 94 = 2.25
the area factor
area of DEF = 40 × 2.25 = 90
square inches
90 square inches
Perimeter is a length, so it takes k, not k². Only the area takes the square, and a
solid would take k³ for its volume.
7
“Note: Figure not drawn to scale” means trust the words, not the picture.
When that line appears, nothing you measure off the drawing is evidence.
Leg XY is labelled 9 and leg XZ is labelled 40, yet they are drawn almost the same
length — ∠Y is drawn at about 63°, while 9, 40 and 41 put it at about 77°. That is
exactly what the note is warning you about, and it is why the
correspondence sentence, not the picture, decides which side pairs with which.
The one thing a figure can still be trusted for is what it states in words: the
right-angle square at X, a tick mark, a point lying on a segment, two segments
meeting at a point. Those are given, not drawn.
✓
Check: does the scale factor hold everywhere?
sides18 ÷ 12 = 1.5k, the one ratio
areas90 ÷ 40 = 2.25and 1.5² = 2.25 ✓
sensible?DEF is the bigger triangleso its area is bigger ✓
Remember the flow
Match the letters · Write the proportion · Square it for area
AA does the proving; the correspondence sentence does the pairing.
Now you try
Practice problem
What is the value of the perimeter?
Triangle ABC is similar to triangle DEF, where A corresponds to D, B corresponds
to E, and C corresponds to F. AB = 10,
BC = 17, AC = 21, and
DE = 25. What is the perimeter of triangle DEF?
DE pairs with AB, so k = 2510 = 2.5. The perimeter of ABC is 10 + 17 + 21 = 48, and perimeter is a length, so it scales by k: 48 × 2.5 = 120. A stops at the perimeter of ABC.B adds the difference once — 48 + (25 − 10) = 63 — instead of scaling all three sides. D uses k² = 6.25, the area factor; only areas take the square.
Hint: you never need the other two sides of DEF separately. Scale the whole
perimeter at once — and check which quantity the question asked for before you
square anything.
