HL Similarity — Definition, Formula & Examples

HL Similarity

Hypotenuse-leg similarity. When two right triangles have corresponding sides with identical ratios as shown below, the triangles are similar.

Two right triangles: △ABC~△DEF with right angles at C and F. AB/DE=AC/DF; hypotenuses and legs are corresponding sides.

See also

Similarity tests for triangles

Key Formula

\frac{\text{Hypotenuse}_1}{\text{Hypotenuse}_2} = \frac{\text{Leg}_1}{\text{Leg}_2}

Where:

$\text{Hypotenuse}_1$ = The hypotenuse of the first right triangle
$\text{Hypotenuse}_2$ = The hypotenuse of the second right triangle
$\text{Leg}_1$ = One leg of the first right triangle
$\text{Leg}_2$ = The corresponding leg of the second right triangle

Worked Example

Problem: Right triangle ABC has a hypotenuse of 10 and a leg of 6. Right triangle DEF has a hypotenuse of 15 and a leg of 9. Are the two triangles similar by HL Similarity?

Step 1: Identify the hypotenuses and the pair of corresponding legs. Hypotenuse₁ = 10, Hypotenuse₂ = 15, Leg₁ = 6, Leg₂ = 9.

Step 2: Compute the ratio of the hypotenuses.

\frac{\text{Hypotenuse}_1}{\text{Hypotenuse}_2} = \frac{10}{15} = \frac{2}{3}

Step 3: Compute the ratio of the corresponding legs.

\frac{\text{Leg}_1}{\text{Leg}_2} = \frac{6}{9} = \frac{2}{3}

Step 4: Compare the two ratios. Both equal 2/3, so the HL Similarity condition is satisfied.

\frac{2}{3} = \frac{2}{3} \checkmark

Answer: Yes, triangle ABC is similar to triangle DEF by HL Similarity, with a scale factor of 2 : 3.

Another Example

This example reverses the problem: instead of testing for similarity, it assumes HL Similarity holds and uses the scale factor to find unknown side lengths, reinforcing the Pythagorean theorem connection.

Problem: Right triangle PQR has a hypotenuse of 13 and one leg of 5. Right triangle STU has a hypotenuse of 26. If the triangles are similar by HL Similarity, find the corresponding leg of triangle STU and the other legs of both triangles.

Step 1: Find the scale factor from the hypotenuses.

\frac{\text{Hypotenuse}_{PQR}}{\text{Hypotenuse}_{STU}} = \frac{13}{26} = \frac{1}{2}

Step 2: Use the scale factor to find the corresponding leg of triangle STU.

\text{Leg}_{STU} = 5 \times 2 = 10

Step 3: Find the other leg of triangle PQR using the Pythagorean theorem.

\text{Other leg}_{PQR} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

Step 4: Apply the scale factor to find the other leg of triangle STU, or use the Pythagorean theorem again as a check.

\text{Other leg}_{STU} = 12 \times 2 = 24 \quad \text{(check: } \sqrt{26^2 - 10^2} = \sqrt{576} = 24 \text{)}

Answer: The corresponding leg of triangle STU is 10, and the remaining legs are 12 (PQR) and 24 (STU). All corresponding sides share the ratio 1 : 2.

Frequently Asked Questions

Why does HL Similarity only work for right triangles?

The right angle is guaranteed in both triangles, so you already know one pair of angles is equal (both 90°). Once the hypotenuse and one leg are in proportion, the Pythagorean theorem forces the remaining legs to be in the same proportion as well. This means all three pairs of sides share a common ratio, which is exactly the condition for similarity. Without the right angle, knowing just two sides are proportional is not enough.

What is the difference between HL Similarity and HL Congruence?

HL Congruence requires the hypotenuse and a leg of one right triangle to be exactly equal to the hypotenuse and a leg of the other. HL Similarity only requires that the ratios of the hypotenuses and the corresponding legs are equal. Congruence is the special case of similarity where the scale factor is 1.

How many sides do you need to check for HL Similarity?

You need to check only two sides: the hypotenuse and one leg from each triangle. Because both triangles are right triangles, the Pythagorean theorem guarantees that if two pairs of corresponding sides are proportional, all three pairs are proportional.

HL Similarity vs. HL Congruence

	HL Similarity	HL Congruence
Applies to	Right triangles only	Right triangles only
What you compare	Ratio of hypotenuses = ratio of a pair of corresponding legs	Hypotenuse₁ = Hypotenuse₂ and Leg₁ = Leg₂
Conclusion	The triangles are similar (same shape, possibly different size)	The triangles are congruent (same shape and same size)
Scale factor	Any positive real number	Exactly 1
Minimum information needed	Hypotenuse and one leg of each triangle	Hypotenuse and one leg of each triangle

Why It Matters

HL Similarity appears frequently in geometry courses when you work with right triangles inscribed in circles, altitude-on-hypotenuse problems, and trigonometric ratio derivations. It is especially useful because you only need two measurements per triangle instead of three, making it one of the most efficient similarity tests. Standardized tests and proofs involving nested right triangles often rely on this criterion.

Common Mistakes

Mistake: Applying HL Similarity to non-right triangles.

Correction: HL Similarity requires both triangles to have a right angle. Without the guaranteed 90° angle, matching just a longest side and one other side in proportion does not prove similarity. For non-right triangles, use AA, SAS Similarity, or SSS Similarity instead.

Mistake: Comparing a leg to the hypotenuse (mixing up which sides correspond).

Correction: Make sure you compare hypotenuse to hypotenuse and leg to the corresponding leg. The hypotenuse is always the side opposite the right angle — the longest side. Comparing a leg of one triangle to the hypotenuse of the other will give incorrect ratios.

Related Terms

Hypotenuse — The longest side of a right triangle
Leg of a Right Triangle — One of the two sides forming the right angle
Similar — Figures with the same shape but possibly different size
Right Triangle — Triangle type required for HL Similarity
Corresponding — Matching sides or angles between similar figures
Similarity Tests for Triangles — All criteria for proving triangle similarity
Triangle — The general polygon HL Similarity applies to
Side of a Polygon — The segments compared in the ratio test