Ratios of Special Triangles — DeltaMath Answers
The maths in this module is two memorised ratios and one multiplication. Almost nobody gets it wrong because they cannot do the arithmetic — they get it wrong because they typed 7.07 into a box that wanted 5√2, or because they multiplied when the side they were solving for needed division. This page covers both triangles, how to identify which side you are looking at, and the exact-form rule that decides whether a correct calculation gets the mark.
- Two triangles only: 45-45-90 in the ratio
1 : 1 : √2, and 30-60-90 in the ratio1 : √3 : 2. - Identify sides by the angle opposite them, never by how long they look in the drawing — DeltaMath diagrams are not to scale.
- The module wants exact radical form. A decimal is marked wrong even when it rounds correctly.
- If a radical ends up on the bottom of a fraction, you have to rationalise before the box will accept it.
What This Module Covers
“Ratios of Special Triangles” deals with exactly two shapes: the 45-45-90 triangle and the 30-60-90 triangle. They are called special because their side lengths sit in a fixed proportion no matter how big the triangle is, so you can find every side from any one side without trigonometry.
DeltaMath gives you a triangle with one side labelled and the angles marked, and asks for one or both of the missing sides. Some versions run it backwards — you get a side and have to identify which special triangle you are in.
It is genuinely one of the quicker modules to master, and that is worth saying because the search volume suggests otherwise. What trips people up is not the geometry. It is the answer box.
The Two Ratios
Memorise these in the order short, long, hypotenuse and most of the module takes care of itself.
| Triangle | Ratio | In words |
|---|---|---|
| 45-45-90 isosceles right triangle | 1 : 1 : √2 | The two legs are equal. The hypotenuse is a leg times √2. |
| 30-60-90 half an equilateral triangle | 1 : √3 : 2 | The hypotenuse is twice the short leg. The long leg is the short leg times √3. |
The 30-60-90 ratio is the one people garble, usually by attaching the √3 to the hypotenuse. It does not go there. The hypotenuse is the clean one — exactly double the short leg — and the awkward √3 belongs to the middle side.
Telling Which Side Is Which
Every error in this module that is not a formatting error is a labelling error. The fix is a rule you can apply without thinking:
A side is identified by the angle directly opposite it. The short leg sits opposite the 30° angle. The long leg sits opposite the 60°. The hypotenuse sits opposite the right angle, always.
Do not judge by appearance. DeltaMath diagrams are frequently drawn out of proportion — a 30-60-90 triangle can be rendered with the two legs looking nearly equal — and the drawing is not the data. The angle marks are the data.
In a 45-45-90 the ambiguity mostly disappears, since the two legs are interchangeable. The only question there is whether the labelled side is a leg or the hypotenuse, and the right-angle square answers it.
Multiply or Divide?
The ratios tell you the relationship, but not the operation. Which one you need depends on whether you are moving toward the bigger side or the smaller one, and this is the second-most-common slip in the module.
Work out which side you have and which you want
Use the opposite-angle rule above. Write both down before doing anything else.
Moving to a bigger side? Multiply
Short leg to hypotenuse in a 30-60-90 means ×2. Leg to hypotenuse in a 45-45-90 means ×√2.
Moving to a smaller side? Divide
Hypotenuse to short leg means ÷2. Hypotenuse to leg in a 45-45-90 means ÷√2 — and that division is what creates the radical-in-the-denominator problem below.
Going between the two legs of a 30-60-90? Route through the short leg
There is no direct clean step from long leg to hypotenuse. Divide by √3 to reach the short leg, then multiply by 2.
Exact Form — Why Your Decimal Is Marked Wrong
This is the single biggest source of frustration on the module, and it is worth being precise about what is happening.
When DeltaMath asks for a side of a special triangle, it is checking your answer against an exact value — something like 5√2 or 8√3. If you calculate 5 × 1.41421... and type 7.07, you have given a rounded approximation of the right answer. The box compares it to the exact value, finds they are not the same number, and marks it wrong.
You did the geometry correctly. You answered a different question than the one asked.
- Leave the radical in.
5√2, not7.07.8√3, not13.86. - Only round when the prompt says to. If it says “round to the nearest tenth,” a decimal is what it wants — and then radical form is the wrong answer.
- Check the answer box itself. DeltaMath supplies a radical template (the √ button) on problems that expect exact form. If that button is there, it is a hint about what the box is checking.
- Simplify the radical.
√8will usually be rejected in favour of2√2.
Rationalising the Denominator
Dividing by a radical — which happens every time you go from a hypotenuse back down to a leg — leaves you with something like 10/√2. That is a correct value, and DeltaMath will usually still reject it, because convention says a radical does not belong in a denominator.
Fixing it takes one move: multiply the top and the bottom by the radical.
| Step | Working |
|---|---|
| Start | 10/√2 |
Multiply top and bottom by √2 | (10 × √2) / (√2 × √2) |
| The bottom becomes a whole number | 10√2 / 2 |
| Simplify | 5√2 |
The reason it works is that √2 × √2 = 2 — a radical times itself clears the root. The value never changed; you multiplied by √2/√2, which is 1. Only the written form changed, and the written form is what is being graded.
Worked Examples
Four problems in the shapes DeltaMath assigns. Your numbers will be different; the routes are not.
| Given | Wanted | Working | Answer |
|---|---|---|---|
| 45-45-90, leg = 6 | Hypotenuse | Leg × √2 | 6√2 |
| 45-45-90, hypotenuse = 10 | Leg | 10 ÷ √2, then rationalise | 5√2 |
| 30-60-90, short leg = 4 | Hypotenuse and long leg | 4 × 2, and 4 × √3 | 8 and 4√3 |
| 30-60-90, hypotenuse = 14 | Short leg and long leg | 14 ÷ 2 = 7, then 7 × √3 | 7 and 7√3 |
Row two is the one to study. The unrationalised answer 10/√2 is numerically identical to 5√2, and only one of them gets the mark.
How Delta Genie Handles Special Triangles
Delta Genie reads the triangle as rendered on your page — the angle marks, the labelled side, the right-angle square — identifies which special triangle it is, and routes to the requested side using the rules above.
The part that matters more than the arithmetic is the entry. Delta Genie writes the answer in the form the box is checking: radical where exact form is expected, simplified, rationalised, and switched to a rounded decimal only when the prompt explicitly asks for one. That formatting layer is where most of the lost marks on this module live, and it is the least interesting part to do by hand.
As on every module, the accuracy and timing controls apply — you set how often it should miss and how long it waits, so a set of special-triangle problems does not come back perfect in forty seconds.
Special Triangles — FAQ
What are the ratios for the two special right triangles?
A 45-45-90 triangle has sides in the ratio 1 : 1 : √2, so both legs are equal and the hypotenuse is a leg times √2. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the hypotenuse is twice the short leg and the long leg is the short leg times √3.
Why does DeltaMath mark my decimal answer wrong?
Because it is checking against an exact value. If the answer is 5√2 and you type 7.07, you have submitted a rounded approximation rather than the value asked for. Leave the radical in unless the prompt explicitly tells you to round.
How do I know which side is the short leg?
By the angle opposite it. The short leg is opposite the 30 degree angle, the long leg is opposite the 60, and the hypotenuse is opposite the right angle. Never judge from the drawing — DeltaMath diagrams are often out of proportion.
What does rationalising the denominator mean?
Removing a radical from the bottom of a fraction. Multiply the numerator and denominator by that radical: 10/√2 becomes 10√2/2, which simplifies to 5√2. The value is unchanged, but DeltaMath expects the rationalised form.
Do I multiply or divide by √3?
Multiply when moving from the short leg to the long leg, and divide when moving from the long leg back to the short leg. To get from the long leg to the hypotenuse, divide by √3 first to reach the short leg, then multiply by 2.
Is there an answer key for ratios of special triangles?
No. The side lengths are re-rolled per student, so the numbers in your version differ from everyone else's. The two ratios on this page are the part that transfers, and they are all you actually need.