7.4 Proportions in right triangles

Proportions in Right Triangles

When you drop an altitude from the right angle of a triangle to the hypotenuse, something powerful happens: the original triangle splits into two smaller triangles, and all three are similar to each other. This similarity creates a set of proportional relationships that let you find unknown lengths without needing every measurement. These relationships center on a concept called the geometric mean.

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Geometric Mean in Right Triangles

The geometric mean of two positive numbers $a$ and $b$ is $\sqrt{a \times b}$. In right triangles, the geometric mean shows up naturally when you draw the altitude from the right angle to the hypotenuse.

That altitude divides the hypotenuse into two segments. The length of the altitude equals the geometric mean of those two segments.

• If the hypotenuse segments are 3 and 12, the altitude is $\sqrt{3 \times 12} = \sqrt{36} = 6$
• If the segments are 5 and 20, the altitude is $\sqrt{5 \times 20} = \sqrt{100} = 10$

This isn't a coincidence. It follows directly from the similar triangles the altitude creates.

Altitude and Hypotenuse Relationships

Three Similar Triangles

Drawing the altitude from the right angle vertex (call it $C$) to the hypotenuse (hitting it at point $D$) produces three similar right triangles:

1. The original triangle $\triangle ABC$
2. The smaller triangle on one side of the altitude ($\triangle ACD$)
3. The smaller triangle on the other side ($\triangle CBD$)

Why are they similar? Each smaller triangle shares one acute angle with the original triangle, and all three triangles have a right angle. Two matching angles is enough to guarantee similarity by AA. Since both smaller triangles are similar to the original, they're also similar to each other. This similarity is the engine behind every proportion in this section.

The Altitude as a Mean Proportional

Because the two smaller triangles are similar to each other, you can write:

$\frac{AD}{CD} = \frac{CD}{DB}$

Here $AD$ and $DB$ are the two hypotenuse segments, and $CD$ is the altitude. Cross-multiplying gives:

$CD^2 = AD \times DB$

So the altitude squared equals the product of the two hypotenuse segments. When a value sits in the middle of a proportion like this (appearing in both the numerator and denominator), it's called the mean proportional between the other two values.

Example: If $AD = 4$ and $CD = 8$, set up $\frac{4}{8} = \frac{8}{DB}$. Cross-multiply to get $4 \times DB = 64$, so $DB = 16$.

Each Leg as a Mean Proportional

The similarity also gives you a proportion for each leg of the original triangle. If $c$ is the full hypotenuse:

• $AC^2 = AD \times c$ (the leg is the geometric mean of its adjacent hypotenuse segment and the whole hypotenuse)
• $BC^2 = DB \times c$

Notice the pattern: each leg is paired with the hypotenuse segment it's adjacent to (the segment it touches), not the far one. This is sometimes called the leg-hypotenuse geometric mean relationship, and it's easy to overlook. Many problems require it, so keep it in your toolkit.

Solving Problems with These Proportions

Here's a reliable approach for problems involving an altitude to the hypotenuse:

1. Draw and label the diagram. Mark the right angle, the altitude, the two hypotenuse segments, and any known lengths.

2. Identify which relationship applies. Ask yourself: do you need the altitude, a hypotenuse segment, or a leg?

   • Need the altitude? Use $CD^2 = AD \times DB$
   • Need a leg? Use $\text{leg}^2 = \text{adjacent segment} \times \text{hypotenuse}$
   • Need a hypotenuse segment? Set up the proportion and solve for the unknown.

3. Set up the equation or proportion and solve.

4. Check with the Pythagorean theorem when possible. Since $a^2 + b^2 = c^2$, you can verify your answer.

Example: In right triangle $\triangle ABC$, the altitude from $C$ to hypotenuse $\overline{AB}$ creates segments $AD = 9$ and $DB = 16$. Find the altitude and both legs.

• Altitude: $CD = \sqrt{9 \times 16} = \sqrt{144} = 12$
• Leg $AC$: $AC = \sqrt{9 \times 25} = \sqrt{225} = 15$ (since the full hypotenuse is $9 + 16 = 25$)
• Leg $BC$: $BC = \sqrt{16 \times 25} = \sqrt{400} = 20$
• Check: $15^2 + 20^2 = 225 + 400 = 625 = 25^2$ ✓

Common Mistakes to Avoid

• Mixing up which segment goes with which leg. Each leg pairs with its adjacent hypotenuse segment. If you accidentally use the wrong segment, your answer will be off.
• Forgetting to add the segments for the full hypotenuse. The leg formula uses the entire hypotenuse, not just one segment. If $AD = 9$ and $DB = 16$, the hypotenuse is 25, not 9 or 16.
• Using the altitude formula when you need the leg formula (or vice versa). Read the problem carefully to identify exactly which length is unknown before picking a formula.

Real-World Applications

Right triangle proportions come up whenever you need to find a distance you can't measure directly.

• Construction and architecture: You can determine the height of a structure using its shadow and the angle of elevation to the sun. A 20 ft pole casting a 15 ft shadow lets you calculate the sun's angle with $\tan(\theta) = \frac{20}{15}$, and then use that angle to find the height of a nearby building from its shadow.
• Navigation and surveying: Measuring the angle to the top of a cliff from a known distance away creates a right triangle. The cliff height follows from the tangent ratio and the measured distance.
• Physics and engineering: Forces acting at an angle break into perpendicular components using right triangle relationships. A force of 100 N at 30° to a surface has a horizontal component of $100\cos(30°)$ and a vertical component of $100\sin(30°)$.