What Is the Length of Side AC of a Triangle? A Complete Guide to Finding It
When you look at a triangle labeled △ABC, the side AC is the segment that connects vertex A to vertex C. In real terms, determining its length is a common problem in geometry, trigonometry, and even real‑world applications such as surveying, architecture, and navigation. Because of that, the method you choose depends on what information you already know about the triangle. Below, we explore the most reliable techniques, walk through step‑by‑step examples, and highlight common mistakes to avoid.
Understanding Triangle Notation
In a triangle, vertices are usually named with capital letters (A, B, C). The side opposite a vertex is named with the lowercase letter of that vertex. Therefore:
- Side a = BC (opposite A)
- Side b = AC (opposite B)
- Side c = AB (opposite C)
When the problem asks for “the length of side AC,” it is asking for side b. Knowing this notation helps you match formulas to the given data That alone is useful..
Methods to Find the Length of Side AC
1. Law of Cosines (Two Sides and the Included Angle)
If you know the lengths of the two sides that meet at vertex B (i.e., AB and BC) and the measure of angle ∠ABC (the angle between those sides), the Law of Cosines gives you AC directly:
[ b^{2}=a^{2}+c^{2}-2ac\cos(\angle B) ]
where
- (a = BC)
- (c = AB)
- (b = AC) (the unknown)
- (\angle B) is the angle at vertex B.
Steps
- Square the known sides (a) and (c).
- Compute (2ac\cos(\angle B)).
- Subtract that product from the sum of the squares.
- Take the square root of the result to obtain (b).
2. Law of Sines (Two Angles and Any Side)
When you have two angles and any one side, you can find the remaining sides using the Law of Sines:
[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
If you know, for example, side AB (c) and its opposite angle C, plus angle B, you can solve for side AC (b):
[ b = \frac{c \cdot \sin B}{\sin C} ]
Steps
- Ensure the angle measures are in the same unit (degrees or radians).
- Plug the known side and its opposite angle into the proportion.
- Multiply and divide as shown to isolate (b).
- Compute the sine values (using a calculator or table) and finish the arithmetic.
3. Right Triangle & Pythagorean Theorem
If △ABC is a right triangle with the right angle at vertex B, then sides AB and BC are the legs, and AC is the hypotenuse. In this case:
[ b^{2}=a^{2}+c^{2} ]
where (a) and (c) are the legs. Simply square the legs, add them, and take the square root But it adds up..
4. Using Area and Height
The area (K) of a triangle can be expressed as:
[ K = \frac{1}{2} \times (\text{base}) \times (\text{height}) ]
If you treat side AC as the base, you need the corresponding height from vertex B to line AC. Rearranging gives:
[ b = \frac{2K}{h} ]
where (h) is the altitude from B onto AC. This method is handy when you know the triangle’s area (perhaps from Heron’s formula) and can measure or compute the height Worth keeping that in mind..
5. Coordinate Geometry
When the vertices are given as points in a Cartesian plane, (A(x_1,y_1)), (B(x_2,y_2)), (C(x_3,y_3)), the distance formula yields the length of any side:
[ b = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} ]
This is essentially the Pythagorean theorem applied to the coordinate differences.
Worked Examples
Example 1 – Law of Cosines
Given: AB = 7 cm, BC = 9 cm, and ∠ABC = 60°. Find AC Most people skip this — try not to..
[ \begin{aligned} b^{2} &= a^{2}+c^{2}-2ac\cos B \ &= 9^{2}+7^{2}-2(9)(7)\cos 60^{\circ} \ &= 81+49-126 \times 0.5 \ &= 130-63 = 67 \ b &= \sqrt{67}\approx 8.19\text{ cm} \end{aligned} ]
Example 2 – Law of Sines
Given: AB = 5 in, ∠C = 30°, ∠B = 45°. Find AC Took long enough..
First find ∠A = 180° – (30°+45°) = 105°. Then:
[ b = \frac{c \sin B}{\sin C} = \frac{5 \sin 45^{\circ}}{\sin 30^{\circ}} = \frac{5 \times \frac{\sqrt{2}}{2}}{0.5} = \frac{5\sqrt{2}}{1} \approx 7.07\text{ in} ]
Example 3 – Right Triangle
Given: Right angle at B, AB = 6 m, BC = 8 m. Find AC.
[ b = \sqrt{6^{2}+8^{2}} = \sqrt{36+64}= \sqrt{100}=10\text{ m} ]
Example 4 – Area and Height
Given: Area = 24 cm², height from B to AC = 4 cm. Find AC.
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