Right Triangles and Trigonometry Pattern - Ratio to Dimension

This pattern gives you a trig ratio (like $\sin P = \dfrac{12}{13}$) and one side length, then asks you to find another side, the perimeter, or the area. The core technique: use the ratio to build a scaled triangle, then extract the measurement you need.

The Scale Factor Method

A trig ratio like $\sin P = \dfrac{12}{13}$ tells you the shape of the triangle — the opposite side and hypotenuse are in a $12:13$ ratio. The actual side lengths are $12k$, $5k$, and $13k$ for some scale factor $k$. Use the known side to find $k$, then compute the target.

Step-by-step: 1. Write the ratio as a fraction: $\sin P = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{12}{13}$ 2. Recognize the Pythagorean triple: $5$-$12$-$13$. The sides are $5k$, $12k$, $13k$ 3. Match the known side to find $k$ 4. Compute the requested measurement

Finding a Missing Side

In $\triangle PQR$, the measure of $\angle Q$ is $90°$ and the length of $PQ$ is $65$ cm. If $\sin P = \dfrac{12}{13}$, what is the length, in centimeters, of segment $QR$?

A) $13$
B) $65$
C) $156$
D) $169$

$\sin P = \dfrac{QR}{PR} = \dfrac{12}{13}$, so the triple is $5$-$12$-$13$.
The sides are: $PQ = 5k$ (adjacent to $P$), $QR = 12k$ (opposite $P$), $PR = 13k$ (hypotenuse).
Given $PQ = 65$: $5k = 65$, so $k = 13$.
$QR = 12k = 12 \times 13 = 156$ cm.
The answer is C. Option A ($13$) is just the scale factor $k$. Option D ($169 = 13k$) gives the hypotenuse, not $QR$.

Gotcha — Which side is which? In $\triangle PQR$ with right angle at $Q$: - $PQ$ is adjacent to angle $P$ (it touches $P$ and the right angle) - $QR$ is opposite angle $P$ (it's across from $P$) - $PR$ is the hypotenuse (opposite the right angle)

Finding the Hypotenuse from Tangent

When given $\tan$ instead of $\sin$ or $\cos$, you have the two legs but not the hypotenuse. Use the Pythagorean theorem (or recognize the triple) to get it.

In right triangle $\triangle PQR$, $\angle Q = 90°$ and $PQ = 90$ inches. If $\tan P = \dfrac{12}{5}$, what is the length, in inches, of the hypotenuse $PR$?

A) $97.5$
B) $18$
C) $234$
D) $216$

$\tan P = \dfrac{QR}{PQ} = \dfrac{12}{5}$, so the triple is $5$-$12$-$13$.
Sides: $PQ = 5k$, $QR = 12k$, $PR = 13k$.
$PQ = 90$: $5k = 90$, so $k = 18$.
$PR = 13k = 13 \times 18 = 234$ inches.
The answer is C. Option D ($216 = 12k$) gives $QR$, not the hypotenuse. Option B ($18$) is just $k$.

Finding the Perimeter

Perimeter problems require finding all three sides, then adding them up.

In $\triangle PQR$, $\angle Q$ is a right angle and $QR = 45$ units. If $\sin P = \dfrac{15}{17}$, what is the perimeter of $\triangle PQR$?

A) $77$
B) $120$
C) $40$
D) $86$

$\sin P = \dfrac{QR}{PR} = \dfrac{15}{17}$, so the triple is $8$-$15$-$17$.
Sides: $PQ = 8k$, $QR = 15k$, $PR = 17k$.
$QR = 45$: $15k = 45$, so $k = 3$.
$PQ = 8(3) = 24$, $QR = 45$, $PR = 17(3) = 51$.
Perimeter $= 24 + 45 + 51 = 120$.
The answer is B.

Using Cosine

In $\triangle PQR$, $\angle Q = 90°$ and $PR = 45$ units. If $\cos Q = \dfrac{8}{17}$... wait — $\angle Q$ is $90°$, so the ratio must reference a different angle. If $\cos P = \dfrac{8}{17}$, then:

$\cos P = \dfrac{PQ}{PR} = \dfrac{8}{17}$. Triple: $8$-$15$-$17$.
$PR = 45$: $17k = 45$, so $k = \dfrac{45}{17}$.
This gives non-integer sides. The perimeter $= (8 + 15 + 17)k = 40 \times \dfrac{45}{17} = \dfrac{1800}{17}$.

Key insight: The scale factor $k$ doesn't have to be an integer. The method works the same way — it's just less "clean."

General Approach for Any Problem in This Pattern

1. Identify the trig ratio. Write it as $\dfrac{\text{one side}}{\text{another side}}$ using SOHCAHTOA
2. Determine the Pythagorean triple from the ratio (or use the Pythagorean theorem to find the third number in the ratio)
3. Label all three sides as multiples of $k$: e.g., $5k$, $12k$, $13k$
4. Use the given side to solve for $k$
5. Compute the target (another side, perimeter, area)

What to Do on Test Day

• Recognize the Pythagorean triple from the ratio. $\dfrac{12}{13}$ → $5$-$12$-$13$. $\dfrac{15}{17}$ → $8$-$15$-$17$. $\dfrac{3}{5}$ → $3$-$4$-$5$
• Label which side is opposite, adjacent, and hypotenuse relative to the given angle before doing anything else
• Scale factor method: All sides scale by the same $k$. Find $k$ from the known side, then multiply to get any other side
• Don't stop at $k$. The most common trap answer is the scale factor itself. The question asks for a side length, not $k$
• Don't confuse the side you found with the side they asked for. If the question asks for $QR$ and you found $PR$, you have the wrong side
• For perimeter: you need all three sides. Find $k$, multiply to get each side, then add
• For area: $\text{Area} = \dfrac{1}{2} \times \text{leg}_1 \times \text{leg}_2$. The hypotenuse is never used directly in the area formula
• If the ratio doesn't match a common triple, use the Pythagorean theorem: if $\sin \theta = \dfrac{a}{c}$, the missing side is $\sqrt{c^2 - a^2}$