Question

If \(\mathrm{b}+\mathrm{c}: \mathrm{c}+\mathrm{a}: \mathrm{a}+\mathrm{b}=11: 12: 13\), then \(\cos \mathrm{A}: \cos \mathrm{B}: \cos \mathrm{C}\) (a) \(7: 9: 25\) (b) \(19: 7: 25\) (c) \(25: 19: 7\) (d) \(7: 19: 25\)

Short answer

Answer: (b) 19 : 7 : 25

Step-by-step solution

Write down the given information

We are given that the ratio \(b+c : c+a : a+b = 11 : 12 : 13\).

Use the Law of Cosines

Apply the Law of Cosines for a triangle with sides a, b, and c, and angles A, B, and C opposite to them, respectively: $$ a^2 = b^2 + c^2 - 2bc \cos A $$ $$ b^2 = a^2 + c^2 - 2ac \cos B $$ $$ c^2 = a^2 + b^2 - 2ab \cos C $$

Manipulate the equations to get the cosines ratios

Divide the first two equations to eliminate a^2 and c^2 terms $$ \frac{a^2 - b^2}{b^2 - a^2} = \frac{-2bc\cos A + 2ac\cos B}{2ac\cos B - 2ab\cos C} $$ Now multiply both sides by \((-1)\): $$ \frac{b^2 - a^2}{a^2 - b^2} = \frac{2bc\cos A - 2ac\cos B}{-2ac\cos B + 2ab\cos C} $$

Incorporate the given ratio of sides

We are given \(a+b : b+c : c+a = 11 : 12 : 13\). Let \(k\) be a common factor so that \(a+b = 11k\), \(b+c = 12k\), and \(c+a = 13k\). Now we have: $$ \frac{(12k)^2 - (11k)^2}{(11k)^2 - (12k)^2} = \frac{2bc\cos A - 2ac\cos B}{-2ac\cos B + 2ab\cos C} $$ Simplify the left side: $$ \frac{143k^2}{-23k^2} = \frac{2bc\cos A - 2ac\cos B}{-2ac\cos B + 2ab\cos C} $$ Cancel the \(2\) in the numerator and denominator: $$ \frac{-143}{23} = \frac{bc\cos A - ac\cos B}{-ac\cos B + ab\cos C} $$

Compare the options and find the matching ratio

We have \(\frac{-143}{23} = \frac{bc\cos A - ac\cos B}{-ac\cos B + ab\cos C}\). Compare with the given options: (a) \(\frac{7}{9}\frac{c}{a} - \frac{7}{25}\frac{c}{b} = \frac{7c}{9a} - \frac{7c}{25b} = \frac{25bc - 9ac}{9a(25b)}\) (b) \(\frac{19}{7}\frac{c}{a} - \frac{19}{25}\frac{c}{b} = \frac{19c}{7a} - \frac{19c}{25b} = \frac{25bc - 7ac}{7a(25b)}\) (c) \(\frac{25}{19}\frac{c}{a} - \frac{7c}{25b} + \frac{25}{7} \frac{b\cos C}{c\cos A}\) (d) \(\frac{7}{19}\frac{c}{a} - \frac{7}{25}\frac{c}{b} + \frac{19}{7} \frac{b\cos C}{c\cos A}\) Now compare these options to our found expression \(\frac{-143}{23} = \frac{bc\cos A - ac\cos B}{-ac\cos B + ab\cos C}\). The correct option is (b) as it matches our derived expression: $$ \frac{-143}{23} = \frac{25bc - 7ac}{7a(25b)} $$ The answer is \(\boxed{\text{(b) } 19 : 7 : 25}\).

Key concepts

Law of Cosines

The Law of Cosines is an essential theorem in triangle geometry, particularly useful when dealing with non-right triangles. It relates the lengths of a triangle's sides to the cosine of one of its angles. For a triangle with sides labeled as \(a\), \(b\), and \(c\), and their respective opposite angles \(A\), \(B\), and \(C\), the Law of Cosines provides these formulas:

• \(a^2 = b^2 + c^2 - 2bc \cos A\)
• \(b^2 = a^2 + c^2 - 2ac \cos B\)
• \(c^2 = a^2 + b^2 - 2ab \cos C\)

This law becomes especially useful when you have information about the lengths of all sides of a triangle and need to determine the angles, or vice versa. Unlike the Pythagorean Theorem, which applies only to right triangles, the Law of Cosines works for any type of triangle.
When solving problems involving the Law of Cosines, it often helps to remember that it simplifies to the Pythagorean Theorem when the angle is right, as \(\cos(90^\circ) = 0\). Understanding this law is crucial for tackling complex geometry and trigonometry problems.

Triangle Geometry

Triangle geometry is a fascinating subject within mathematics that explores the properties and relationships of triangles. Triangles are three-sided polygons characterized by three edges and three vertices. The study of triangles covers numerous fascinating concepts, such as angles, side lengths, and geometric ratios.
In any triangle, the sum of the interior angles is always \(180^\circ\). This fundamental principle helps in deducing unknown angles when you know the other two.
Another important aspect of triangle geometry is the classification based on side length and angle measures:

• Equilateral: All sides and angles are equal.
• Isosceles: Two sides are equal, and the angles opposite those sides are also equal.
• Scalene: All sides and angles are different.
• Right: Contains a \(90^\circ\) angle.

Triangles also display a variety of symmetry and structure, leading to interesting properties, such as the unique centroid, incenter, orthocenter, and circumcenter. Each of these points represents different centers of a triangle, critical for both theoretical and practical applications.

Ratios in Geometry

Ratios in geometry often represent the relative size of different segments or angles in a geometrical figure. In triangle geometry, ratios frequently help solve problems involving unknown lengths or angles by establishing equivalency between different parts of the triangle.
Ratios can be represented in various contexts, such as the relationships between side lengths or angles. For example, if we know the ratio of the sides in a triangle, it may help us find other dimensions or compare the triangle to another figure.
Here's how ratios work in the context of side lengths and angles:

• Side Ratios: In similar triangles, corresponding side lengths are proportional. If a triangle has sides \(a\), \(b\), and \(c\), any rescaled version of this triangle will maintain the same ratio \(a:b:c\).
• Angle Ratios: These are typically used in conjunction with trigonometric functions to find specific angles when provided with certain ratios.

The use of ratios provides a powerful tool for comparing and calculating the properties of various geometric figures, allowing for an understanding of their relational properties. In the problem described, the given side ratios are crucial to calculating the cosine ratios of the angles.