Question

If the angles of a triangle are in the ratio \(3: 4: 5\), then the ratio of the corresponding sides is (a) \(2: \sqrt{2}: 3\) (b) \(2: \sqrt{6}: \sqrt{3}+1\) (c) \(\sqrt{3}: 2: \sqrt{5}\) (d) \(\sqrt{2}: \sqrt{6}: \sqrt{3}+1\)

Short answer

Answer: The ratio of the sides of the triangle is \(a:b:c = 2 : \sqrt{6} : \sqrt{3} + 1\).

Step-by-step solution

Calculate the actual angles of the triangle from the given ratio

We know the ratio of the angles is \(3:4:5\). Let the angles be \(3x\), \(4x\), and \(5x\) degrees. We also know the sum of angles of a triangle is 180 degrees. Therefore, we can create an equation: \(3x + 4x + 5x = 180\)

Solve for x

Combine the terms and solve the equation for the value of x: \(12x = 180\) Now divide by 12: \(x = 15\)

Find the actual angle values

Now substitute the value of x into the expressions for the angles to find their actual values: Angle 1: \(3x = 3(15) = 45\) degrees Angle 2: \(4x = 4(15) = 60\) degrees Angle 3: \(5x = 5(15) = 75\) degrees

Apply the sine rule to find the ratio of sides

Let the ratio of the corresponding sides be \(a:b:c\). According to the sine rule: \( \frac{a}{\sin(45)} = \frac{b}{\sin(60)} = \frac{c}{\sin(75)}\) To find the ratio of the sides, we need to find the value that relates these sides. Let's start with \(\frac{a}{\sin(45)}\).

Calculate the ratios for side a, b, and c

a) \(\frac{a}{\sin(45)} = \frac{b}{\sin(60)} \Rightarrow a = b\cdot\frac{\sin(45)}{\sin(60)}\) In the case of \(\sin(45)\), we get \(\frac{\sqrt{2}}{2}\); and for \(\sin(60)\), we get \(\frac{\sqrt{3}}{2}\). Thus, \(a = b\cdot\frac{\sqrt{2}}{\sqrt{3}}\) b) We can do the same for side c: \(\frac{a}{\sin(45)} = \frac{c}{\sin(75)} \Rightarrow a = c\cdot\frac{\sin(45)}{\sin(75)}\) It's important to notice that \(\sin(75)\) can also be written as \(\sin(30+45)\). Using the sine addition formula, we get: \(\sin(75) = \sin(30+45) = \sin(30)\cos(45) + \cos(30)\sin(45) = \frac{1}{2}\cdot\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\) So, \(a = c\cdot\frac{\sqrt{2}}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} = c\cdot\frac{4\sqrt{2}}{\sqrt{2} + \sqrt{6}}\)

Choose the correct answer

Comparing the side ratios with the options given, we see that the correct answer is option (b), which is \(a:b:c = 2 : \sqrt{6} : \sqrt{3} + 1\)

Key concepts

Sine Rule in Triangles

The sine rule is a powerful relationship that helps us find unknown sides or angles in any triangle, not just right-angled ones. The sine rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides and angles in a triangle.

Mathematically, this can be expressed as:
\[\begin{equation}\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)},\end{equation}\]where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite these sides respectively. To successfully apply the sine rule, one must know at least one side and its opposite angle, or two angles and one side. The exercise improvement advice here is clarity. Complex operations like converting a given angle ratio to actual angles, and then using the sine rule to find side ratios require clear, step-by-step explanation. Understanding how to use the given angle ratio to find actual angles using the angle sum property is the first step for the sine rule application.

Angle Sum Property of Triangles

A fundamental property of all triangles is that the sum of the interior angles always equals 180 degrees. This knowledge is extremely crucial when solving problems involving unknown angles. Whether the triangle is scalene, isosceles, or equilateral, this rule always holds true.

Using this principle, we can often find the measure of one or more unknown angles when given some angles or ratios. In the given problem, the ratio of the angles is used alongside the angle sum property to establish the actual angles which are key to solving the remaining parts of the problem. It is important to highlight that without correctly applying the angle sum property, the sine rule would not be applicable. Ensuring students comprehend the angle sum property is pivotal in many geometric problems and enhances their problem-solving toolset.

Solving Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. The foundational step in solving these equations is to remember the fundamental trigonometric identities and ratios for key angles, which often come into play.

For example, familiar values such as \(\sin(30^\circ) = \frac{1}{2}\), \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), and \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) are essential when dealing with angles commonly encountered in problems. Furthermore, in more advanced cases, the use of trigonometric formulas such as the sine addition formula, as demonstrated in the solution, becomes necessary.

Understanding these equations is imperative when applying the sine rule in the context of a triangle problem. Students need to become comfortable with these equations to utilize the sine rule effectively. Thus, in educational content, elucidating the process of solving for these values and their practical application in the context of the problem enhances learning outcomes and student confidence.