Lesson 52: Using Two Special Right Triangles

New Concept

These triangles, called special right triangles, are the $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle and the $45^\circ\text{-}45^\circ\text{-}90^\circ$ triangle.

Why it matters

Mastering these triangles isn't just about geometry; it's about recognizing that fixed relationships can unlock unknown values, a core principle in all of algebra. This skill—leveraging known patterns to solve for unknowns—is fundamental for everything from physics to advanced function analysis.

What’s next

Next, you'll use these fixed side ratios to find missing lengths in both $45^\circ\text{-}45^\circ\text{-}90^\circ$ and $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangles.

In a $45°-45°-90°$ right triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg multiplied by $\sqrt{2}$. The side ratios follow a $1-1-\sqrt{2}$ pattern.

If a leg is $7$ cm, the hypotenuse is $7\sqrt{2}$ cm.
If the hypotenuse is $10\sqrt{2}$ feet, each leg is $\frac{10\sqrt{2}}{\sqrt{2}} = 10$ feet.
If a leg measures $9$, the hypotenuse is $9\sqrt{2}$.

Think of this triangle as a perfect square sliced in half diagonally. This is why its two legs are always equal! To find the hypotenuse (the long side), just take the length of a leg and multiply it by $\sqrt{2}$. It’s a super shortcut that saves you from doing the full Pythagorean theorem every single time.

In a $30°-60°-90°$ triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is the shorter leg's length times $\sqrt{3}$. The side ratios follow a $1-\sqrt{3}-2$ pattern.

If the shorter leg is $5$, the hypotenuse is $2 \times 5 = 10$ and the longer leg is $5\sqrt{3}$.
If the hypotenuse is $14$, the shorter leg is $\frac{14}{2} = 7$ and the longer leg is $7\sqrt{3}$.
If the longer leg is $6\sqrt{3}$, the shorter leg is $6$ and the hypotenuse is $12$.

This triangle is an equilateral triangle neatly cut in half. The shortest leg (opposite the $30°$ angle) is your key. The hypotenuse is always double the short leg, and the longer leg (opposite the $60°$ angle) is the short leg multiplied by $\sqrt{3}$. This simple relationship makes finding side lengths a total breeze!

Verify using the Pythagorean Theorem: Show that the hypotenuse of a $45°-45°-90°$ triangle is the length of a leg times $\sqrt{2}$.

Let both legs equal $x$. Then $x^2 + x^2 = c^2$, which simplifies to $2x^2 = c^2$.
Taking the square root gives $\sqrt{2x^2} = \sqrt{c^2}$, so the hypotenuse $c$ equals $x\sqrt{2}$.
For a triangle with legs of 6: $6^2 + 6^2 = 36 + 36 = 72$. So, $c = \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

Ever wonder if these shortcuts are just math magic? They're actually the Pythagorean theorem in a cool disguise! For a $45°-45°-90°$ triangle, the legs (a and b) are equal. Plugging this into $a^2 + b^2 = c^2$ becomes $a^2 + a^2 = c^2$. This proves our shortcut is based on solid mathematical logic!