Ostatnia aktualizacja: 19 sierpnia 2026
Kalkulator Similar Right Triangles
Twórcy
Dharmendra SinghRecenzenci
Twórcy
Dharmendra SinghRecenzenci
Quick Answer
The similar right triangles calculator uses the altitude-to-hypotenuse theorems for a right triangle split into segments p and q. It computes the full hypotenuse c = p + q, the altitude h = √(pq), and the legs a = √(pc) and b = √(qc), which come directly from the similarity of the three right triangles in the diagram.
Use the similar right triangles calculator when an altitude splits the hypotenuse into p and q, then apply c equals p plus q, h equals root pq, and each leg equals the square root of its segment times the whole hypotenuse.
Kluczowe Wnioski
- Drawing the altitude to the hypotenuse creates two smaller triangles similar to the original.
- The altitude is the geometric mean of the hypotenuse segments: h² = pq.
- Each leg is the geometric mean of the whole hypotenuse and its adjacent segment.
- The whole hypotenuse is simply p + q.
- Checking a² + b² = c² is a reliable final verification step.
Twórcy
Dharmendra SinghRecenzenci
Twórcy
Dharmendra SinghRecenzenci
Wzór
h² = pq, a² = p(p+q), b² = q(p+q)
Gdzie:
- p=First hypotenuse segment(units)
- q=Second hypotenuse segment(units)
- h=Altitude to the hypotenuse(units)
- a, b=Leg lengths of the original right triangle(units)
Rozwiązane przykłady
Segments 4 and 9
Use the geometric-mean relationships for a clean benchmark example.
- 1Add the segments to get the full hypotenuse: 4 + 9 = 13.
- 2Use h² = pq, so h = √36 = 6.
- 3Use a² = p(p+q) = 4 × 13 and b² = q(p+q) = 9 × 13.
Segments 5 and 20
A larger imbalance between the two segments changes both legs but keeps the same theorems.
- 1The hypotenuse is 25.
- 2The altitude is √(5 × 20) = 10.
- 3The legs are √125 and √500.
Segments 3 and 12
This example shows another exact altitude with irrational legs.
- 1The hypotenuse is 3 + 12 = 15.
- 2The altitude is √36 = 6.
- 3The legs are √45 and √180.
Wprowadzenie
The Similar Right Triangles Calculator uses the classic altitude-to-hypotenuse theorems from right-triangle geometry. When you draw the altitude from the right angle to the hypotenuse, the original triangle splits into two smaller right triangles, and all three triangles are similar. That similarity creates geometric-mean relationships that let you recover the altitude, the whole hypotenuse, and both legs from the two hypotenuse segments alone. This page is intentionally focused on that standard theorem setup rather than on generic triangle similarity problems.
What the similar-right-triangle diagram means
In a right triangle, draw the altitude from the right angle to the hypotenuse. That one segment splits the original triangle into two smaller right triangles. Each small triangle shares an acute angle with the original, so all three triangles are similar. The labels p and q describe the two pieces of the hypotenuse after the split.
The original triangle is right-angled.
The altitude lands on the hypotenuse.
The hypotenuse is divided into p and q.
All three resulting triangles are similar.
Geometric-mean relationships
The core relationships are h² = pq, a² = p(p+q), and b² = q(p+q). The altitude is the geometric mean of the two hypotenuse segments, and each leg is the geometric mean of the full hypotenuse and the adjacent segment. These equalities are powerful because they replace angle chasing with direct algebra.
Altitude squared equals the product pq.
Each leg squared equals adjacent segment times whole hypotenuse.
The whole hypotenuse is p + q.
Square roots recover the actual lengths.
Why similarity creates these formulas
Because the triangles are similar, corresponding side ratios are equal. Rearranging those proportions produces the geometric-mean theorems. This is a good example of how similarity turns a diagram into an algebra system: the altitude and legs are not guessed separately, but derived from the shared ratio structure of the three triangles.
Similarity creates matching angle structures.
Matching ratios turn into product formulas.
The geometric mean appears naturally from proportion rearrangement.
The theorems work for any right triangle with the altitude drawn to the hypotenuse.
Worked example with segments 4 and 9
When p = 4 and q = 9, the whole hypotenuse is 13. The altitude satisfies h² = 4 × 9 = 36, so h = 6. The leg next to p satisfies a² = 4 × 13 = 52, giving a ≈ 7.211103. The other leg satisfies b² = 9 × 13 = 117, giving b ≈ 10.816654. Those values also agree with the Pythagorean theorem for the original triangle.
Add p and q first.
Use h² = pq for the altitude.
Use a² = pc and b² = qc for the legs.
Check the original triangle with a² + b² = c².
Common mistakes
A common mistake is mixing up the segment next to a leg with the whole hypotenuse. Another is forgetting to add p and q before using the leg formulas. Users also sometimes square where they should take a square root, or assume the altitude equals one of the segments even though it is actually the geometric mean of both segments together.
Do not use p or q alone as the whole hypotenuse.
Do not skip the square root when solving for a length.
Do not swap h² = pq with the leg formulas.
Do not ignore which segment is adjacent to which leg.
Where these theorems are used
These formulas appear most often in geometry courses, but they also matter anywhere scaled right-triangle relationships are used for proofs, drafting, or computational geometry. The calculator is especially useful in study settings because the intermediate outputs mirror the exact quantities shown in textbook diagrams, making it easier to compare your work with a labeled figure.
Geometry homework and exam review.
Proof checking with similar triangles.
Drafting and diagram verification.
Computational geometry sanity checks.
When to use this calculator
Use this page when your problem specifically involves a right triangle cut by an altitude to the hypotenuse. If the problem gives ordinary side lengths or acute angles without that altitude split, a general right-triangle calculator may be better. This page is best when the diagram labels p and q or asks for the geometric-mean relationships directly.
Use it when the altitude-to-hypotenuse theorem appears.
Use it when p and q are given directly.
Switch to a general right-triangle tool for other setups.
Keep the original diagram nearby while interpreting the outputs.
Manual method versus calculator use
These formulas are compact enough to solve by hand, so the calculator works best as a verification tool or as a time-saver when decimals appear. A strong workflow is add the segments, compute the altitude, then compute both legs and finally check the Pythagorean theorem. The page mirrors that order so each output has a clear geometric meaning.
Solve c = p + q first.
Compute the altitude before the legs.
Use the outputs in the same order the theorem is taught.
Confirm that the larger leg sits next to the larger segment.
Karta Szybkiego Odniesienia
Similar right triangles quick reference
Szybkie odniesienie • Kalkulator Similar Right Triangles
c = p + q, h = √(pq), a = √(pc), b = √(qc).Prawidłowy zakres: Use positive hypotenuse segments p and q greater than 0.
Typowe Wartości
⚠ Uwaga
- •Do not forget to add p and q before using the leg formulas.
- •Do not skip the square root when turning squared relationships into lengths.
- •Keep the segment labels tied to the correct leg outputs.
- •Use a different calculator if the problem does not involve the altitude to the hypotenuse.
Wskazówki Pro
- →Find the whole hypotenuse first so the leg formulas are easier to read.
- →Use h² = pq as a quick check because it is often the simplest computation.
- →Compare the larger segment with the larger leg to catch label swaps.
- →Finish by checking the Pythagorean theorem on the original triangle.
FAQ
Why are the triangles called similar?
The original right triangle and the two smaller triangles formed by the altitude all share matching angle measures. Equal angles imply similar triangles.
What is the formula for the altitude to the hypotenuse?
If the altitude splits the hypotenuse into segments p and q, then h² = pq, so h = √(pq).
How do I find the whole hypotenuse?
Add the two hypotenuse segments: c = p + q. That total is then used in both leg formulas.
What are the leg formulas in this setup?
If c = p + q, then the leg adjacent to p satisfies a² = pc and the leg adjacent to q satisfies b² = qc.
Can the two segments be equal?
Yes. If p = q, the altitude lands at the midpoint of the hypotenuse and the original triangle becomes an isosceles right triangle.
How can I check that the outputs make sense?
After finding the two legs and the whole hypotenuse, verify that a² + b² = c². That confirms the values match the original right triangle.
When should I use a different triangle calculator?
Use a general right-triangle calculator when your problem is based on side lengths, angles, or trigonometric ratios without the special altitude-to-hypotenuse split.