Given Two Sides Of A Triangle Find The Third

Finding the length of a missing side in a triangle is one of the most fundamental skills in geometry, yet the method changes drastically depending on the type of triangle and the information available. While the Pythagorean theorem is the first tool most students learn, it applies exclusively to right triangles. For the vast majority of triangles—oblique triangles—you must rely on the Law of Cosines or the Law of Sines, often requiring an angle measurement to calculate a precise length. Understanding which rule governs your specific scenario is the key to solving these problems accurately That's the part that actually makes a difference..

The Critical Role of the Triangle Type

Before reaching for a formula, you must classify the triangle. The relationship between the two known sides and the unknown third side is not fixed; it is governed by the angles between them Simple, but easy to overlook..

Right Triangles: The Pythagorean Theorem

If you are given two sides of a right triangle, the solution is straightforward. The Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides (legs) It's one of those things that adds up. No workaround needed..

$a^2 + b^2 = c^2$

• Finding the Hypotenuse: If you know the two legs ($a$ and $b$), square them, add the results, and take the square root. $c = \sqrt{a^2 + b^2}$
• Finding a Leg: If you know the hypotenuse ($c$) and one leg ($a$), subtract the square of the known leg from the square of the hypotenuse, then take the square root. $b = \sqrt{c^2 - a^2}$

Example: A right triangle has legs of 6 cm and 8 cm. The third side (hypotenuse) is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm.

Oblique Triangles: Why Two Sides Are Not Enough

If the triangle is not a right triangle (acute or obtuse), knowing only two side lengths makes it mathematically impossible to determine a single, unique length for the third side.

Imagine two rigid sticks of known lengths connected by a hinge at one end. Practically speaking, you can open or close that hinge, changing the angle between the sticks. That's why as the angle changes, the distance between the free ends (the third side) changes continuously. Without knowing the angle—or at least one other angle in the triangle—the third side could be almost any length within a specific range That's the whole idea..

Counterintuitive, but true Not complicated — just consistent..

This brings us to the Triangle Inequality Theorem, which defines the possible range for the third side Easy to understand, harder to ignore..

The Triangle Inequality Theorem: Defining the Range

Even without angles, you can establish the boundaries within which the third side must fall. For any triangle with sides $a$, $b$, and $c$:

$|a - b| < c < a + b$

• The Upper Bound: The third side must be shorter than the sum of the other two sides. If it were equal to or longer, the two known sides could not meet to close the shape.
• The Lower Bound: The third side must be longer than the absolute difference of the other two sides. If it were equal to or shorter, the longer side would swallow the shorter one, preventing a triangle from forming.

Example: Given sides of length 7 and 12 Small thing, real impact..

• Upper bound: $7 + 12 = 19$. The third side ${content}lt; 19$.
• Lower bound: $|12 - 7| = 5$. The third side ${content}gt; 5$.
• Possible Range: $5 < x < 19$.

This theorem is essential for checking the validity of answers in standardized tests and real-world engineering constraints The details matter here..

Solving Oblique Triangles: The Law of Cosines

When you have two sides and the included angle (the angle between the two known sides), you have a Side-Angle-Side (SAS) scenario. This is the perfect setup for the Law of Cosines. It works for any triangle—right, acute, or obtuse—and is essentially a generalized version of the Pythagorean theorem.

The Formula

For a triangle with sides $a$, $b$, $c$ and angles $A$, $B$, $C$ (where side $a$ is opposite angle $A$, etc.):

$c^2 = a^2 + b^2 - 2ab\cos(C)$

To find the third side ($c$), you need:

1. Side $a$
2. Side $b$

Step-by-Step Calculation

1. Square the two known sides ($a^2$ and $b^2$).
2. Multiply the two known sides together, multiply by 2, and multiply by the cosine of the included angle ($2ab\cos(C)$).
3. Subtract the result from step 2 from the sum of the squares in step 1.
4. Take the square root of the result.

Worked Example: Given side $a = 10$, side $b = 14$, and included angle $C = 60^\circ$.

1. $a^2 = 100$, $b^2 = 196$. Sum $= 296$.
2. $2ab\cos(C) = 2(10)(14)\cos(60^\circ) = 280(0.5) = 140$.
3. $c^2 = 296 - 140 = 156$.
4. $c = \sqrt{156} \approx 12.49$.

Special Case Verification: If $C = 90^\circ$, $\cos(90^\circ) = 0$. The formula reduces to $c^2 = a^2 + b^2$, proving the Pythagorean theorem is a subset of the Law of Cosines.

Solving Oblique Triangles: The Law of Sines

If you are given two sides and a non-included angle (Side-Side-Angle or SSA), you must use the Law of Sines. This scenario is famously known as the Ambiguous Case because it can yield zero, one, or two possible triangles.

The Formula

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

The Ambiguous Case (SSA) Workflow

Suppose you know side $a$, side $b$, and angle $A$ (opposite side $a$). You want to find side $c$.

1. Find Angle $B$: Use the Law of Sines: $\sin(B) = \frac{b \sin(A)}{a}$.
2. Analyze the Result:
   • If $\sin(B) > 1$: No triangle exists. The side $a$ is too short to reach the baseline.
   • If $\sin(B) = 1$: One right triangle exists ($B = 90^\circ$).
   • If $\sin(B) < 1$: Two possible angles exist for $B$: $B_1 = \arcsin(\dots)$ (acute) and $B_2 = 180^\circ - B_1$ (obtuse).
3. Check Validity: For each potential $B$, calculate $C = 180^\circ - A - B$. If $C > 0$, that triangle is valid.