One equation, and the hypotenuse is always the one sitting alone.
What is the value of the distance?
In the xy-plane, point A has coordinates (−3, 2) and point B
has coordinates (9, −3). What is the distance between A and B?
The horizontal gap is 9 − (−3) = 12 and the vertical gap is 2 − (−3) = 5, so the distance is √12² + 5² = √169 = 13. D adds the two gaps, 12 + 5 — the answer you get by walking around the corner instead of cutting across. A subtracts them, 12 − 5. B subtracts the squares, √144 − 25, which treats the 12 as the hypotenuse when it is a leg.
1
a² + b² = c², and c is always the side opposite the right angle.
The two legs share a side of the equation; the hypotenuse sits alone on the other.
a² + b² = c²
a and b are the legs, in either order. c is the hypotenuse — every time.
How to spot the hypotenuse without thinking
It is the side that does not touch the right-angle mark, and it is always the
longest side of the triangle. In a word problem it is whatever spans the gap — the
ladder, the ramp, the diagonal, the straight-line distance — while the legs are the
two perpendicular directions you would otherwise travel.
2
Hypotenuse missing, add. Leg missing, subtract.
Same equation both times; all that changes is which letter you are solving for.
looking for the hypotenuse
c = √a² + b²
the answer is bigger than either leg
looking for a leg
a = √c² − b²
the answer is smaller than the hypotenuse
A right triangle has a hypotenuse of 41 and one leg of 9. What is the length of the other leg?
9² + b² = 41²
41 is the hypotenuse, so it goes alone
81 + b² = 1,681
square both givens
b² = 1,600
subtract 81 from both sides
b = 40
and 40 < 41, as a leg must be
The one check that catches the commonest error
Adding here would give √41² + 9² ≈ 41.97, a leg longer than the
hypotenuse it belongs to. Whenever your answer for a leg comes out bigger than the
longest side, you added when you should have subtracted.
3
Three triples are worth knowing on sight.
Recognising one turns thirty seconds of squaring into none at all.
Triple
Why it works
Multiples you will actually meet
3, 4, 5
9 + 16 = 25
6-8-10, 9-12-15, 12-16-20, 30-40-50
5, 12, 13
25 + 144 = 169
10-24-26, 15-36-39, 25-60-65
8, 15, 17
64 + 225 = 289
16-30-34, 24-45-51
7, 24, 25
49 + 576 = 625
14-48-50
Multiply every side of a triple by the same number and it stays a triple, because
each square is multiplied by that number squared. The opening problem was 5-12-13.
A triple is three sides, not any two numbers
Seeing a 5 and a 13 tells you 12 only if the 13 is the hypotenuse. If 5 and 13 are
both legs, the hypotenuse is √194 ≈ 13.93 and nothing is clean. Always check which
side the big number is before you reach for a memorised third value.
4
On the xy-plane, the distance between two points is a hypotenuse.
Go across, then up, and the straight line back is the third side of a right triangle.
horizontal leg = 9 − (−3) = 12
the x-values, subtracted
vertical leg = 2 − (−3) = 5
the y-values, subtracted
d² = 12² + 5² = 144 + 25 = 169
the distance is the hypotenuse, so add
d = 13
a 5-12-13 triangle on the grid
AB = 13
The distance formula in a textbook, d = √(x₂ − x₁)² + (y₂ − y₁)², is this same
triangle written out. There is nothing extra to learn — and a subtraction that comes
out negative does not matter, because it is squared immediately.
5
A rectangle's diagonal cuts it into two right triangles.
In rectangle PQRS, PS = 30 and PQ = 16. What is the length of diagonal PR?
PQ² + QR² = PR²
angle Q is a right angle, so PR is the hypotenuse
16² + 30² = 256 + 900 = 1,156
QR equals PS, the opposite side
PR = √1,156 = 34
8-15-17 doubled
PR = 34
The SAT usually runs this backwards: it gives you the diagonal and one side and
asks for the perimeter or the area. Nothing changes — find the missing side by
subtracting the squares, then answer the question that was actually asked, which is
rarely the side itself.
6
In an isosceles triangle, the height splits the base in half.
Triangle ABC has AB = AC = 26 and BC = 20. What is the height from A to BC?
half the base = 20 ÷ 2 = 10
the height lands exactly in the middle
h² + 10² = 26²
26 is the slanted side, so it is the hypotenuse
h² = 676 − 100 = 576
subtract, because h is a leg
h = 24
and the area is ½(20)(24) = 240
Half the base, not the whole base
The right triangle you are working in has legs h and 10, not h and 20. Using 20
gives √676 − 400 ≈ 16.6, and the answer to the area question that follows is then
wrong by a long way. The split is only in half because the two slanted sides are
equal — in a scalene triangle the foot of the height is somewhere else entirely.
7
When the answer is not a whole number, leave it under the root.
A right triangle has legs of 6 and 9. What is the length of the hypotenuse?
c² = 6² + 9² = 36 + 81 = 117
no triple here
c = √117 = √9 × 13
pull out the largest perfect square
c = 3√13 ≈ 10.82
between 9 and 15, so it is plausible
Where Desmos earns its place
The choices are often written as radicals, so type both: sqrt(117) and
3*sqrt(13) both return 10.8166…, which confirms the simplification in
a second. If instead the question is student-produced and asks for a decimal, the
calculator hands you the digits to type, and you keep at least four of them before
rounding.
A useful bound while you work: the hypotenuse is always longer than either leg but
shorter than the two legs added together. Here that is 9 < c < 15, which rules
out most wrong choices before you calculate anything.
✓
Check: rebuild the triangle from your answer.
the legs12² + 5² = 144 + 25= 169
the distance13² = 169a match ✓
sensible?12 < 13 < 17longer than a leg, shorter than both ✓
Remember the flow
Find the right angle · Name the hypotenuse · Add or subtract
3-4-5, 5-12-13, 8-15-17 — and every multiple of them.
Now you try
Practice problem
What is the value of the area?
In triangle ABC, AB = AC = 20 and
BC = 24. What is the area of triangle ABC?
The height from A splits BC into two 12s, so h² = 20² − 12² = 400 − 144 = 256 and h = 16. Area = ½(24)(16) = 192. C uses the slanted side 20 as the height, ½(24)(20) — but 20 is the hypotenuse, so the height has to be shorter. D forgets the ½.A uses the half-base 12 as the whole base, ½(12)(16).
Hint: draw the height from A down to BC before anything else. It makes two right
triangles, and the leg you need comes from subtracting the squares — and the triangle
it makes is a familiar triple scaled up.
