Question

which of these triangles can you use the law of cosines to solve for a missing side?

Explanation:

The Law of Cosines is used when we know two sides and the included angle (SAS) or all three sides (SSS) of a triangle to find a missing side or angle. Let's analyze each triangle:

Step 1: Analyze the first triangle (right triangle with 35° angle)

This is a right triangle. For right triangles, the Pythagorean theorem (\(a^2 + b^2 = c^2\)) or trigonometric ratios (sine, cosine, tangent) are more appropriate. The Law of Cosines can be used, but it's not the most straightforward here, and it doesn't fit the typical SAS or SSS case for when we specifically choose Law of Cosines over simpler methods. Also, we don't have two sides and the included angle in a way that Law of Cosines is necessary here.

Step 2: Analyze the second triangle (with sides 12, 8 and included angle 138°)

Here, we know two sides (12 and 8) and the included angle (138°) between them. This is a classic SAS case, which is suitable for applying the Law of Cosines. The Law of Cosines formula for a triangle with sides \(a\), \(b\), included angle \(C\), and side \(c\) opposite angle \(C\) is \(c^2=a^2 + b^2-2ab\cos(C)\). In this triangle, if we want to find side \(x\), we can use the Law of Cosines with \(a = 12\), \(b=8\) and \(C = 138^{\circ}\).

Step 3: Analyze the third triangle (equilateral triangle with 60° angles)

This is an equilateral triangle (all angles 60°), so all sides are equal. We can find the side length using properties of equilateral triangles or Law of Sines, but since we don't have two sides and an included angle (or three sides) in a way that we need Law of Cosines, and also, if we try to use Law of Cosines here, it would be redundant because we know the triangle is equilateral. Also, we don't have two sides and an included angle given (we know angles but not two sides and the included angle to apply Law of Cosines for finding a side).

Answer:

The triangle with sides 12, 8 and included angle \(138^{\circ}\) (the second triangle) can be solved for the missing side using the Law of Cosines.