Right Triangles and Trigonometry Pattern - Special Triangles

This pattern tests your knowledge of the two special right triangles — the $45°$-$45°$-$90°$ and $30°$-$60°$-$90°$ triangles — and their fixed side-length ratios. These ratios let you find missing sides without using a calculator.

 

The 45-45-90 Triangle (Isosceles Right Triangle)

$\text{Sides: } x : x : x\sqrt{2}$

The two legs are equal (both $x$), and the hypotenuse is $x\sqrt{2}$.

• Given a leg, find the hypotenuse: multiply by $\sqrt{2}$
• Given the hypotenuse, find a leg: divide by $\sqrt{2}$ (equivalently, multiply by $\dfrac{\sqrt{2}}{2}$)

This triangle appears whenever you see: an isosceles right triangle, a square's diagonal, or a $45°$ angle in a right triangle.

A piece of fabric is cut in the shape of an isosceles right triangle. The perimeter is $20 + 20\sqrt{2}$ cm. What is the length, in centimeters, of the hypotenuse?

A) $10\sqrt{2}$
B) $20$
C) $10$
D) $20\sqrt{2}$

Let each leg $= x$. Then hypotenuse $= x\sqrt{2}$.
Perimeter $= x + x + x\sqrt{2} = 2x + x\sqrt{2} = x(2 + \sqrt{2})$.
Set equal: $x(2 + \sqrt{2}) = 20 + 20\sqrt{2}$.
Factor the right side: $20(1 + \sqrt{2})$. Hmm — let's try $x = 10\sqrt{2}$: perimeter $= 2(10\sqrt{2}) + 10\sqrt{2} \cdot \sqrt{2} = 20\sqrt{2} + 20 = 20 + 20\sqrt{2}$. ✓
Hypotenuse $= x\sqrt{2} = 10\sqrt{2} \cdot \sqrt{2} = 20$.
The answer is B.

Gotcha — Square diagonals: A square with side $s$ has diagonal $s\sqrt{2}$. The diagonal splits the square into two $45°$-$45°$-$90°$ triangles.

 

The 30-60-90 Triangle

$\text{Sides: } x : x\sqrt{3} : 2x$

• The side opposite $30°$ is the shortest ($x$)
• The side opposite $60°$ is $x\sqrt{3}$ (the medium side)
• The side opposite $90°$ is $2x$ (the hypotenuse, always the longest)

Critical: The hypotenuse is exactly twice the shortest side.

In the triangle shown, $\angle R = 30°$ and $\angle Q = 90°$. The hypotenuse $PR = 22$. What is the value of $z$ (the side opposite $30°$)?

A) $44$
B) $22$
C) $11\sqrt{3}$
D) $11$

The side opposite $30°$ is the shortest side: $x = \dfrac{\text{hypotenuse}}{2} = \dfrac{22}{2} = 11$.
The answer is D. Option C ($11\sqrt{3}$) is the side opposite $60°$, not $30°$. Option A ($44$) goes the wrong way — it doubles instead of halving.

In right triangle $PQR$, $\angle P = 60°$ and $PQ = 12$. What is the value of $y$ (the hypotenuse)?

A) $12$
B) $12\sqrt{3}$
C) $24$
D) $60$

$\angle P = 60°$ and $\angle Q = 90°$, so $\angle R = 30°$.
Side $PQ$ is adjacent to $60°$ and opposite $30°$, so $PQ$ is the shortest side: $PQ = x = 12$.
Hypotenuse $= 2x = 24$.
The answer is C. Option B ($12\sqrt{3}$) gives the side opposite $60°$, not the hypotenuse.

Gotcha — Which side is which? Always identify which angle each side is opposite to: - Opposite $30°$ → shortest → $x$ - Opposite $60°$ → medium → $x\sqrt{3}$ - Opposite $90°$ → hypotenuse → $2x$

Many errors come from confusing the $60°$ side with the hypotenuse. The hypotenuse is $2x$ (no radical), and the $60°$ side is $x\sqrt{3}$.

 

Working Backward from Perimeter

The perimeter of a $30°$-$60°$-$90°$ triangle is $30 + 30\sqrt{3}$ cm. What is the length of the hypotenuse?

A) $20$
B) $30$
C) $20\sqrt{3}$
D) $10\sqrt{3}$

Perimeter $= x + x\sqrt{3} + 2x = 3x + x\sqrt{3} = x(3 + \sqrt{3})$.
$x(3 + \sqrt{3}) = 30 + 30\sqrt{3} = 30(1 + \sqrt{3})$.
Solve: $x = \dfrac{30(1 + \sqrt{3})}{3 + \sqrt{3}}$. Rationalize by multiplying by $\dfrac{3 - \sqrt{3}}{3 - \sqrt{3}}$:
$x = \dfrac{30(1 + \sqrt{3})(3 - \sqrt{3})}{9 - 3} = \dfrac{30(3 - \sqrt{3} + 3\sqrt{3} - 3)}{6} = \dfrac{30(2\sqrt{3})}{6} = 10\sqrt{3}$.
Hypotenuse $= 2x = 20\sqrt{3}$.
The answer is C.

 

Equilateral Triangles: The 30-60-90 Connection

An equilateral triangle with side $s$ has height $= \dfrac{s\sqrt{3}}{2}$. Why? Drop an altitude from any vertex — it bisects the opposite side and creates two $30°$-$60°$-$90°$ triangles.

• Area of equilateral triangle: $A = \dfrac{s^2 \sqrt{3}}{4}$

The area of an equilateral triangle is $144\sqrt{3}$ square inches. The perimeter is $6k$ inches. What is the value of $k$?

$\dfrac{s^2 \sqrt{3}}{4} = 144\sqrt{3}$
$s^2 = 576$
$s = 24$
Perimeter $= 3s = 72 = 6k$, so $k = 12$.
The answer is $12$.

 

Circumscribed and Inscribed Circles with Equilateral Triangles

For an equilateral triangle with side $s$: - Circumradius (center to vertex): $R = \dfrac{s}{\sqrt{3}} = \dfrac{s\sqrt{3}}{3}$. Equivalently, $R = \dfrac{2}{3} \times \text{height}$. - Inradius (center to midpoint of side): $r = \dfrac{s}{2\sqrt{3}} = \dfrac{s\sqrt{3}}{6}$. Equivalently, $r = \dfrac{1}{3} \times \text{height}$.

A circle is circumscribed about an equilateral triangle. The altitude of the triangle is $9\sqrt{3}$ cm. What is the radius of the circle?

A) $3\sqrt{3}$
B) $18$
C) $6\sqrt{3}$
D) $9\sqrt{3}$

The circumradius is $\dfrac{2}{3}$ of the height: $R = \dfrac{2}{3} \times 9\sqrt{3} = 6\sqrt{3}$.
The answer is C. Option A ($3\sqrt{3}$) gives $\dfrac{1}{3}$ of the height — that's the inradius, not the circumradius.

Gotcha — Circumradius vs. inradius: The circumradius ($\dfrac{2}{3}$ of height) goes from center to a vertex. The inradius ($\dfrac{1}{3}$ of height) goes from center to the midpoint of a side. Students frequently confuse the two.

 

What to Do on Test Day

• Memorize the two ratios:
• $45°$-$45°$-$90°$: $x : x : x\sqrt{2}$
• $30°$-$60°$-$90°$: $x : x\sqrt{3} : 2x$ (opposite $30°$, $60°$, $90°$)
• The hypotenuse of a 30-60-90 is $2x$ (no radical). The $\sqrt{3}$ goes with the medium side, not the hypotenuse. This is the most common mix-up
• Equilateral triangle height: $h = \dfrac{s\sqrt{3}}{2}$. Area: $A = \dfrac{s^2\sqrt{3}}{4}$
• Square diagonal: $d = s\sqrt{2}$. The diagonal creates two $45°$-$45°$-$90°$ triangles
• Circumradius $= \dfrac{2}{3} \times$ height (from center to vertex). Inradius $= \dfrac{1}{3} \times$ height (from center to side midpoint)
• Perimeter problems: Write the perimeter as $x(1 + 1 + \sqrt{2})$ or $x(1 + \sqrt{3} + 2)$, set equal to the given expression, and solve for $x$
• When the answer has radicals ($\sqrt{2}$ or $\sqrt{3}$): for MCQ, match the radical form in the answer choices — don't convert to decimals. For SPR, you may need to compute the decimal (e.g., $5\sqrt{3} \approx 8.66$) since you cannot type radicals in the answer choices