Two shapes whose sides are locked into a ratio, so one length gives you all three.
What is the value of the length?
In triangle XYZ, the measure of angle Y is 90°, the measure of angle X is 45°, and XZ = 14. What is the length of YZ?
Angle Y is the right angle, so XZ = 14 is the hypotenuse, and the other two angles are both 45°. In a 45-45-90 the hypotenuse is a leg times √2, so the leg is 14 ÷ √2 = 7√2 ≈ 9.9. D multiplies by √2 instead of dividing, giving a leg of 19.8 — longer than the hypotenuse it belongs to. A halves the hypotenuse, which is the 30-60-90 rule used in the wrong triangle. C uses √3, the other triangle's radical.
Both figures below are drawn at their true proportions, so you can see what the ratios mean.
1 : 1 : √2
half a square, cut corner to corner
1 : √3 : 2
half an equilateral triangle
These are not approximations. Any triangle with those angles has exactly those side ratios, whatever its size — so a single given length unlocks the other two.
The ratios are written in the order of the angles they face: smallest angle, smallest side.
| Angle | The side opposite it | Its share of the ratio |
|---|---|---|
| 30° | the short leg | 1 — half the hypotenuse |
| 60° | the long leg | √3 — the short leg times √3 |
| 90° | the hypotenuse | 2 — twice the short leg |
| 45° and 45° | the two legs | 1 and 1 — they are equal |
| 90° (in a 45-45-90) | the hypotenuse | √2 — a leg times √2 |
In 1 : √3 : 2 the √3 sits in the middle because it is the middle-length side — the leg facing 60°. The hypotenuse is the 2. Students who remember “30-60-90 means √3” and attach the √3 to the longest side get every one of these questions wrong by a factor of 2√3. Read the ratio as a sentence: short leg 1, long leg √3, hypotenuse 2.
Forward is easy. The SAT prefers to hand you the hypotenuse and make you come back.
leg 9 → hypotenuse 9√2 ≈ 12.73
forward: multiply
hypotenuse 14 → leg = 14 ÷ √2
backwards: divide, from the opening problem
14√2 × √2√2 = 14√22
clear the root from the bottom
leg = 7√2 ≈ 9.9
shorter than 14, as a leg must be
Both legs are 7√2, because the two 45° angles face equal sides. So the opening triangle is 7√2, 7√2, 14 — and Pythagoras agrees: 98 + 98 = 196 = 14². ✓
Short leg is the hub. Hypotenuse ÷ 2, or long leg ÷ √3 — then build out from there.
short leg
x
long leg
x√3
hypotenuse
2x
given the hypotenuse 18: x = 18 ÷ 2 = 9
halve it — no radical involved
long leg = 9√3 ≈ 15.59
between 9 and 18, so it is plausible
given the long leg 12: x = 12√3 = 4√3
12√3 ÷ 3 = 4√3 ≈ 6.93
hypotenuse = 2(4√3) = 8√3
double the short leg, never the long one
Doubling the long leg 12 to get a hypotenuse of 24 is the classic slip. Only the short leg doubles. Routing through the short leg every single time costs one extra line and removes the error completely.
The stem will say “square”, not “45-45-90”. Drawing the diagonal is the whole step.
square of side 7 → diagonal 7√2 ≈ 9.9
the two sides are the equal legs
square of diagonal 6√2 → side 6
divide by √2 and the radical cancels
that square's area = 6² = 36
and the question usually wants the area
The same triangle turns up in a 45° ramp, in the distance from a corner of a chessboard to the opposite corner, and any time a stem pairs the word “square” with the word “diagonal”. Each acute angle of the halved square is 45° because the diagonal bisects the 90° corner.
The height cuts the 60° apex in half and cuts the base in half, making two 30-60-90s.
half the base = 10 ÷ 2 = 5
the short leg, facing the 30°
h = 5√3 ≈ 8.66
the long leg, facing the 60°
area = ½(10)(5√3) = 25√3
≈ 43.3 square units
In general, an equilateral triangle of side s has height s√32 and area s²√34. Both come straight out of this one picture, so neither is worth memorising separately.
√2 ≈ 1.414 and √3 ≈ 1.732 — two numbers worth carrying around.
When the four choices are radicals that look alike — 6√3, 12, 6√2, 4√3 — type each
one and compare: 6*sqrt(3) is 10.39, 6*sqrt(2) is 8.49,
4*sqrt(3) is 6.93. Against a hypotenuse of 12 only one of them can be
the long leg. The calculator also settles a simplification instantly:
12sqrt(3) and 4*sqrt(3) both read 6.928…, so they are the
same number.
What is the value of the perimeter?
The length of a diagonal of a square is 10√2 inches. What is the perimeter, in inches, of the square?
The diagonal is a leg times √2, so the side is 10√2 ÷ √2 = 10, and the perimeter is 4(10) = 40. D treats the diagonal as a side and gives 4(10√2). A stops at twice the side, 2(10), forgetting that a square has four of them. B halves the diagonal to 5√2 and multiplies by 4 — the 30-60-90 rule again, in the wrong triangle.
Hint: the diagonal already carries a √2, which is the clue that the side will not. Find one side first, then read the question again — it asks for the perimeter, not the side and not the area.
