Finding the Value of Each Variable in a Triangle
Triangles are among the most fundamental shapes in geometry, and solving for unknown sides or angles is a skill that appears in everything from basic homework problems to engineering designs. Whether you are dealing with a right triangle, an obtuse triangle, or a set of similar figures, the process relies on a handful of core principles: the angle‑sum theorem, the Pythagorean theorem, the Law of Sines, and the Law of Cosines. This guide walks you through each method, shows when to apply it, and provides step‑by‑step examples so you can confidently determine the value of any variable in a triangle.
1. Core Principles You’ll Need
| Principle | When to Use | Formula / Rule |
|---|---|---|
| Angle‑Sum Theorem | You know two interior angles and need the third. Worth adding: | ( \angle A + \angle B + \angle C = 180^\circ ) |
| Pythagorean Theorem | The triangle is right‑angled and you need a missing side. Which means | ( a^2 + b^2 = c^2 ) (where (c) is the hypotenuse) |
| Law of Sines | You have an angle‑side pair (AAS or ASA) or two angles and a non‑included side (SSA). | ( \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} ) |
| Law of Cosines | You have two sides and the included angle (SAS) or all three sides (SSS) and need an angle or the third side. | ( c^2 = a^2 + b^2 - 2ab\cos C ) |
| Similar Triangles | Triangles share the same shape; corresponding sides are proportional. |
Understanding which tool fits the given information is the first step toward solving for the unknown variable(s).
2. Step‑by‑Step Problem‑Solving Framework
- Identify What Is Known – List all given side lengths and angle measures.
- Determine the Triangle Type – Right, acute, obtuse, or ambiguous (SSA).
- Choose the Appropriate Theorem – Use the table above to match known data with a formula.
- Set Up the Equation – Plug known values into the selected formula, leaving the unknown as the variable.
- Solve Algebraically – Isolate the variable, perform any necessary inverse trigonometric functions, and simplify.
- Check for Reasonableness – Verify that angles sum to (180^\circ) and that side lengths obey the triangle inequality.
- State the Answer – Include units (if any) and round to the requested precision.
3. Worked Examples
Example 1: Right Triangle – Using the Pythagorean Theorem
Problem: In right triangle ( \triangle XYZ ), ( \angle Y = 90^\circ ), ( XY = 6 ) cm, and ( YZ = 8 ) cm. Find ( XZ ).
Solution:
- Known: legs (XY = 6), (YZ = 8); hypotenuse unknown.
- Apply Pythagorean theorem: ( XZ^2 = XY^2 + YZ^2 ).
- Compute: ( XZ^2 = 6^2 + 8^2 = 36 + 64 = 100 ).
- Take square root: ( XZ = \sqrt{100} = 10 ) cm.
Answer: ( XZ = 10 ) cm.
Example 2: Obtuse Triangle – Using the Law of Cosines
Problem: In ( \triangle ABC ), ( AB = 7 ) m, ( AC = 9 ) m, and ( \angle A = 120^\circ ). Find side ( BC ) Simple, but easy to overlook. Nothing fancy..
Solution:
- Known: two sides and the included angle (SAS).
- Law of Cosines: ( BC^2 = AB^2 + AC^2 - 2(AB)(AC)\cos A ).
- Plug values: ( BC^2 = 7^2 + 9^2 - 2(7)(9)\cos 120^\circ ).
- Recall ( \cos 120^\circ = -\tfrac{1}{2} ).
- Compute: ( BC^2 = 49 + 81 - 2(7)(9)(-0.5) = 130 + 63 = 193 ).
- ( BC = \sqrt{193} \approx 13.89 ) m.
Answer: ( BC \approx 13.9 ) m.
Example 3: Using the Law of Sines (AAS)
Problem: In ( \triangle DEF ), ( \angle D = 45^\circ ), ( \angle E = 70^\circ ), and side ( DF = 12 ) in. Find side ( EF ).
Solution:
- Find the third angle: ( \angle F = 180^\circ - 45^\circ - 70^\circ = 65^\circ ).
- We have angle‑side pair ( \angle D ) opposite side ( EF ) (unknown) and ( \angle F ) opposite side ( DF = 12 ).
- Law of Sines: ( \dfrac{EF}{\sin D} = \dfrac{DF}{\sin F} ).
- Substitute: ( \dfrac{EF}{\sin 45^\circ} = \dfrac{12}{\sin 65^\circ} ).
- Solve for ( EF ): ( EF = 12 \times \dfrac{\sin 45^\circ}{\sin 65^\circ} ).
- Using approximate values: ( \sin 45^\circ \approx 0.7071 ), ( \sin 65^\circ \approx 0.9063 ).
- ( EF \approx 12 \times \dfrac{0.7071}{0.9063} \approx 12 \times 0.780 = 9.36 ) in.
Answer: ( EF \approx 9.4 ) in.
Example 4: Similar Triangles
Problem: Triangle ( \triangle PQR ) is similar to ( \triangle STU ). In ( \triangle PQR ), ( PQ = 5 ) cm, ( QR = 7 ) cm, and ( PR = 9 ) cm Not complicated — just consistent. Simple as that..
Example 4: Similar Triangles
Problem:
( \triangle PQR ) is similar to ( \triangle STU ). In ( \triangle PQR ), ( PQ = 5 ) cm
Example 5: Area of a Triangle Using Heron’s Formula
Problem: Determine the area of a triangle whose side lengths are 7 cm, 8 cm, and 9 cm.
Solution:
- Compute the semiperimeter (s = \frac{7+8+9}{2}=12) cm.
- Apply Heron’s formula:
[ \text{Area}= \sqrt{s,(s-7),(s-8),(s-9)} = \sqrt{12\cdot5\cdot4\cdot3} = \sqrt{720}\approx 26.83\text{ cm}^2. ] - Round to two decimal places as requested.
Answer: (26.83) cm².
Example 6: Law of Cosines for an SAS Triangle
Problem: In triangle (LMN), (LM = 5) cm, (LN = 7) cm, and (\angle L = 45^\circ). Find side (MN).
Solution:
- Use the Law of Cosines:
[ MN^{2}=LM^{2}+LMSY
