🔷Honors Geometry Unit 8 Review

Special right triangles have fixed angle measures and predictable side length ratios, which means you can find missing sides without reaching for the Pythagorean theorem every time. The two families you need to know are 45-45-90 and 30-60-90 triangles. They show up constantly in geometry problems, standardized tests, and real-world applications like construction and navigation.

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Properties of 45-45-90 Triangles

A 45-45-90 triangle is an isosceles right triangle: two 45° angles and one 90° angle. Because the two acute angles are equal, the two legs are always congruent.

Side length ratio: $x : x : x\sqrt{2}$

The two legs each have length $x$ .
The hypotenuse has length $x\sqrt{2}$ .

Finding a missing side:

Given a leg, multiply by $\sqrt{2}$ to get the hypotenuse. If a leg is 5, the hypotenuse is $5\sqrt{2}$ .
Given the hypotenuse, divide by $\sqrt{2}$ to get each leg. If the hypotenuse is $2\sqrt{2}$ , each leg is $\frac{2\sqrt{2}}{\sqrt{2}} = 2$ .

Rationalizing the denominator: You'll often see $\frac{h}{\sqrt{2}}$ rewritten as $\frac{h\sqrt{2}}{2}$ . Both forms are correct, but your teacher may prefer the rationalized version. To rationalize, multiply the numerator and denominator by $\sqrt{2}$ .

Where they appear: The diagonal of a square splits it into two 45-45-90 triangles. So if a square has side length $s$ , its diagonal is $s\sqrt{2}$ . This comes up any time you cut a square along its diagonal, such as in tile layouts or when calculating the distance between opposite corners of a square room.

Properties of 45-45-90 triangles, Right Triangle Trigonometry | Precalculus

Properties of 30-60-90 Triangles

A 30-60-90 triangle has three sides of different lengths, each tied to a specific angle. The key is correctly identifying which side is opposite which angle.

Short leg (opposite the 30° angle): length $x$
Long leg (opposite the 60° angle): length $x\sqrt{3}$
Hypotenuse (opposite the 90° angle): length $2x$

Side length ratio: $x : x\sqrt{3} : 2x$

A common mistake is mixing up the long leg and the hypotenuse. Remember: the hypotenuse is always $2x$ (a clean multiple of the short leg), while the long leg is $x\sqrt{3}$ (the one with the radical). The hypotenuse must be the longest side, and since $2 > \sqrt{3} \approx 1.73$ , the order from shortest to longest is always short leg, long leg, hypotenuse.

Finding a missing side:

Given the short leg ( $x$ ): multiply by $\sqrt{3}$ for the long leg, multiply by 2 for the hypotenuse. If the short leg is 4, the long leg is $4\sqrt{3}$ and the hypotenuse is 8.
Given the hypotenuse: divide by 2 to get the short leg, then multiply that by $\sqrt{3}$ for the long leg. If the hypotenuse is 10, the short leg is 5 and the long leg is $5\sqrt{3}$ .
Given the long leg: divide by $\sqrt{3}$ to get the short leg, then double the short leg for the hypotenuse. If the long leg is $6\sqrt{3}$ , the short leg is 6 and the hypotenuse is 12.

When the long leg doesn't have a $\sqrt{3}$ in it: If the long leg is just a plain number like 9, you still divide by $\sqrt{3}$ . The short leg would be $\frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}$ , and the hypotenuse would be $6\sqrt{3}$ . Don't panic when the given value doesn't "look clean." Just follow the ratio.

Where they come from: Drop an altitude from any vertex of an equilateral triangle to the opposite side, and you get two congruent 30-60-90 triangles. They also appear inside regular hexagons (which are made of six equilateral triangles) and in problems involving 30° or 60° angles of elevation.

Properties of 45-45-90 triangles, Right Triangle Trigonometry · Algebra and Trigonometry

Solving Real-World Problems with Special Triangles

When a problem gives you specific angle measures, check whether you're dealing with a special right triangle before doing anything else. Here's a general approach:

Identify the triangle type. Look at the angle measures. If you see 45°-45°-90° or 30°-60°-90°, you can use the ratios.
Label the sides. Figure out which side is given and which role it plays (leg, short leg, long leg, or hypotenuse). Drawing and labeling a diagram is worth the 10 seconds it takes.
Apply the ratio to find the missing side(s). Always start from the short leg in a 30-60-90, or from a leg in a 45-45-90. If you're given a different side, work backward to the reference side first.
Check your answer with the Pythagorean theorem ( $a^2 + b^2 = c^2$ ). Your calculated sides should satisfy this equation.
Interpret the result. The answer should be a positive value in appropriate units. Leave answers in simplified radical form unless the problem asks for a decimal approximation.

Example problems:

A square room measures 12 ft on each side. The diagonal (for a support beam) is $12\sqrt{2} \approx 16.97$ ft. That's a 45-45-90 triangle.
A flagpole casts a shadow, and the angle of elevation from the tip of the shadow to the top of the pole is 60°. If the shadow is 20 ft long, the shadow is the short leg (opposite 30°), so the flagpole height (opposite 60°) is $20\sqrt{3} \approx 34.64$ ft.
A ramp needs to reach a loading dock 3 ft high at a 30° angle. The height is opposite the 30° angle (short leg), so the ramp length (hypotenuse) is $2 \times 3 = 6$ ft, and the horizontal distance along the ground (long leg) is $3\sqrt{3} \approx 5.20$ ft.

🔷Honors Geometry Unit 8 Review