Mean Proportional

You face a right triangle with the altitude drawn from the right angle to the hypotenuse. The teacher calls the altitude a mean proportional. The room goes quiet. The formula is simple, but the reason it works is not obvious until you see the similar triangles hiding inside the figure.

The mean proportional between two numbers a and b is the number x that satisfies a/x = x/b. Multiply across, and x² = ab, so x = √(ab). That is the entire definition. It is also called the geometric mean, because a rectangle with sides a and b has the same area as a square with side x. The name matters less than the relationship: the mean proportional is the middle term of a continued proportion, where the ratio of the first to the second equals the ratio of the second to the third.

In a geometry class, the mean proportional shows up in two places: the altitude to the hypotenuse and the legs of a right triangle. Both come from the same similarity argument. Once you see the pattern, the formulas are just bookkeeping.

The mean proportional of two positive numbers a and b is x such that a/x = x/b. Solving gives x = √(ab). For example, the mean proportional between 4 and 9 is √(36) = 6, because 4/6 = 6/9. The order of a and b does not change the result, since multiplication commutes.

The geometric mean is the same thing under a different name. In a set of n positive numbers, the geometric mean is the nth root of the product. For two numbers, that is the square root of the product. For three numbers, it is the cube root. The mean proportional is always the geometric mean of two terms, but the geometric mean can extend to any number of terms. A proportion definition is an equation stating that two ratios are equal, and the mean proportional is the special case where one ratio is repeated.

The formula fails for negative numbers in the real number system, because the square root of a negative product is not real. If you need a mean proportional between two negative numbers, the geometric mean is undefined. That is not a limitation of the formula; it is a property of the real numbers.

A continued proportion is a sequence where the ratio between consecutive terms is constant. Written as a:b = b:c, the middle term b is the mean proportional between a and c. For example, 2, 4, 8 is a continued proportion because 2/4 = 4/8 = 1/2. Here b = 4 is the mean proportional between 2 and 8.

To find the mean proportional in a continued proportion, set up the equation a/b = b/c and solve for b. Cross-multiply to get b² = ac, then take the square root. This works whether you are given two numbers and asked for the middle term, or given the middle term and one end and asked for the other end.

Continued proportion is not the same as compound proportion. A compound proportion involves multiple variables, such as a is proportional to b and inversely to c. Continued proportion is a single chain of equal ratios. The rule of three, a historical method for solving proportions with three known terms and one unknown, applies to continued proportion when you know three of the four values. The unitary method, finding the value of one unit first, is another way to approach the same problem, but it is more natural for direct proportion word problems than for continued proportion.

In a right triangle, draw the altitude from the right angle to the hypotenuse. This creates two smaller right triangles, each similar to the original and to each other. The similarity condition is AA: each small triangle shares one acute angle with the original and has the right angle. This is the geometric mean right triangle theorem, and it is the reason the altitude and the legs are mean proportionals.

The altitude on hypotenuse is the mean proportional between the two segments it creates on the hypotenuse.For example, with hypotenuse 10 and segments 4 and 6, the altitude is √(4·6) = √24 ≈ 4.899. The CK-12 Geometry text, Section 8.2 on Right Triangle Similarity, works this exact example.

Each leg of the original triangle is the mean proportional between the hypotenuse and the adjacent segment.These are called the leg projection theorems.

The altitude theorem and the leg theorems are not separate facts. They all follow from the same set of similar triangles. If you can draw the altitude and label the segments, you can write the proportions directly.

Example 1: Mean Proportional Between Two Numbers

Find the mean proportional between 4 and 16.Check: 4/8 = 1/2 and 8/16 = 1/2. The answer is 8. This is the geometric mean of 4 and 16.

For three terms in continued proportion, the middle term is still the square root of the product of the outer terms.The sequence 2, 4, 8 has a constant ratio of 2.

Example 2: Altitude in a Right Triangle

A right triangle has a hypotenuse of length 13. The altitude to the hypotenuse divides it into segments of 9 and 4. Find the altitude. Using h² = p·q, h² = 9·4 = 36, so h = 6. The altitude is 6 units. The leg adjacent to the segment of 9 has length a = √(13·9) = √117 ≈ 10.82. The other leg has length b = √(13·4) = √52 ≈ 7.21.

Example 3: Leg as Mean Proportional

In a right triangle, the hypotenuse is 25 and one segment is 16. The leg adjacent to that segment is √(25·16) = √400 = 20. The other segment is 25 - 16 = 9, and the other leg is √(25·9) = √225 = 15. The altitude is √(16·9) = √144 = 12. All three geometric mean relationships hold.

These examples cover the two main uses: finding a missing term in a continued proportion and finding lengths in a right triangle. The same formula, x = √(ab), appears in both. The context tells you which a and b to use.

Quantity	Formula	Example (c = 10, p = 4, q = 6)
Altitude h	h² = p·q	h = √(4·6) = √24 ≈ 4.899
Leg a (adjacent to p)	a² = p·c	a = √(4·10) = √40 ≈ 6.325
Leg b (adjacent to q)	b² = q·c	b = √(6·10) = √60 ≈ 7.746
Hypotenuse segments sum	p + q = c	4 + 6 = 10

Avoid Denominator Errors

The most common error is putting the wrong segment in the denominator. In the formula a/x = x/b, the unknown x must be the middle term. If you swap a and b, you get the same answer for the mean proportional because multiplication commutes, but the proportion itself is written incorrectly. Check the order of the ratios.

Use the Right Method

Another error is applying direct proportion where the relationship is not direct. The mean proportional only works when the ratio is constant. If you are solving a word problem about speed or cost, the unitary method may be more appropriate than setting up a continued proportion.

Label the Triangle First

When using the altitude theorem, be careful with the segments. The altitude is the mean proportional between the two segments of the hypotenuse, not between a segment and the whole hypotenuse. That is the leg theorem. Mixing these up is the most common mistake in geometry homework. Draw the triangle and label the parts before writing any equation.

Keep Radicals Exact

Finally, do not round intermediate steps. Keep the radical exact until the end. If the answer is √24, leave it as √24 or simplify to 2√6, not 4.9, unless the problem asks for a decimal. Rounding early introduces errors that compound.

The mean proportional is not a one-off formula. It is the backbone of similarity in right triangles, which appears in every geometry textbook. […] Once you see the mean proportional in the triangle, you can solve for any missing length without memorizing three separate formulas.

The same idea carries into trigonometry. […] If you can solve a/x = x/b, you can solve any proportion.

This topic suits students who like a formula they can apply directly and who benefit from drawing a diagram. It does not suit someone who wants a quick answer without understanding why the similar triangles work. The altitude theorem is easy to misapply if you do not draw the figure first. If you are that student, the worked examples above are the minimum practice you need before a test.