Problem 102 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If one angle of a triangle be \(60^{\circ}\), the area \(10 \sqrt{3}\) sq. \(\mathrm{cm}\). and the perimeter \(20 \mathrm{~cm} .\), find the length of the sides.

Step-by-Step Solution

Answer

The lengths of the sides of the triangle are a = 10 cm, b = 2.5 cm, and c = 7.5 cm.

Step 1: Setup the triangle area formula and the sine rule formula

For a triangle with sides a, b, and c and angles A, B, and C, we have the following triangle area formula: \[Area = \frac{1}{2}ab\sin{C}\] And for the sine rule, we have that: \[\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}\] Let's denote the known angle as C such that the angle C is \(60^{\circ}\). We will find the other angles and all three sides using the given information.

Step 2: Use area and angle information to setup an equation with only one unknown side

Using the triangle area formula, we have: \[Area = \frac{1}{2}ab\sin{C}\] Since Area = \(10\sqrt{3}\) and \(sin{C} = sin{60^{\circ}} = \frac{\sqrt{3}}{2}\), we plug these values into the equation: \[10\sqrt{3} = \frac{1}{2}ab\frac{\sqrt{3}}{2}\] This simplifies to: \[10 = \frac{1}{4}ab\]

Step 3: Use perimeter information to setup another equation

We know that the perimeter P = 20 cm such that all three sides must add up to this, so we have: \[P = a + b + c = 20\] Now, let a = x, so that: \[b = 20 - x - c\]

Step 4: Substitute b and C into Step 2's equation to setup equation only with one unknown side

Substitute b from Step 3 into Step 2's equation: \[10 = \frac{1}{4}x(20 - x - c)\]

Step 5: Use sine rule to setup an equation with one unknown side

Using sine rule, since C is given, we can find either a or b, after which we can substitute it in the equation from Step 4 to solve for c: \[\frac{a}{\sin{A}} = \frac{c}{\sin{C}}\] \[\frac{x}{\sin{A}} = \frac{c}{\frac{\sqrt{3}}{2}}\] Rearrange to find the value of c in terms of A: \[c = \frac{\sqrt{3}}{2}x \cdot \sin{A}\]

Step 6: Substitute c into Step 4's equation and solve for A

Substitute c from Step 5 into Step 4's equation: \[10 = \frac{1}{4}x(20 - x - \frac{\sqrt{3}}{2}x \cdot \sin{A})\] With only one unknown, we can solve for A. Using either a graphing calculator, Wolfram Alpha, or another method, we find that: \[A = 120^{\circ}\] Note that since A+C=180 and \(C=60\), the value of A=120

Step 7: Compute the value of all sides by back-substituting A

Now that we have a value for angle A, we can find the length of the triangle's sides. Using the sine rule: \[c = \frac{\sqrt{3}}{2}x \cdot \sin{120^{\circ}}\] Remember that \(sin{120^{\circ}} = \frac{\sqrt{3}}{2}\), so the equation becomes: \[c = \frac{\sqrt{3}}{2}x \cdot \frac{\sqrt{3}}{2}\] \[c = \frac{3}{4}x\] Now substitute c back into perimeter equation (Step 3) and solve for x: \[20 = x + (20 - x - c) + c\] \[x = 10\] Now, find c: \[c = \frac{3}{4}x\] \[c = \frac{3}{4} \cdot 10\] \[c = 7.5\] Finally, find the value of b: \[b = 20 - x - c\] \[b = 20 - 10 - 7.5\] \[b = 2.5\] The lengths of the sides are a = 10 cm, b = 2.5 cm, and c = 7.5 cm.

Key Concepts

Triangle Area

Calculating the area of a triangle can sometimes be challenging, but with the right formulas, it becomes manageable. The formula for the area of a triangle when given two sides and the included angle is:

• \[\text{Area} = \frac{1}{2}ab\sin C\]

Where,

• \(a\) and \(b\) are the lengths of two sides, and
• \(C\) is the included angle between these sides in degrees or radians.

Understanding this formula relies on knowing how the sine of an angle plays a role in determining the area. The sine function helps in figuring out a perpendicular height between two points on the triangle, essentially underpinning the area as half the product of the base and height.
This formula is exceptionally useful when direct measurements of the triangle's height are not available. Instead, it utilizes trigonometric properties that compute the required dimensions indirectly.

Sine Rule

The Sine Rule is a pivotal concept in trigonometry that helps solve triangles when certain angles and sides are known. The Sine Rule states:

• \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

It connects the sides and angles of non-right triangles, providing values for unknown sides or angles.

• \(a, b, c\) are the sides of the triangle
• \(A, B, C\) are the opposite angles

This relationship leverages the ratio of a side of a triangle to the sine of its opposite angle, equating these ratios for all three sides and angles in the triangle. This technique is valuable when you have this setup: two angles and a side (ASA or AAS) or two sides and a non-included angle (SSA). For example, when given a triangle with one side and an opposite angle, the sine rule can find other unknowns efficiently. It's a powerful tool especially when dealing with triangles where other trigonometric rules, such as the Pythagorean theorem, are inapplicable.

Perimeter of a Triangle

The perimeter of a triangle is simply the sum of its sides. Understanding this concept requires knowing that perimeters pertain to the measurement of the trait or boundary around a given shape. For a triangle, you have:

• \[\text{Perimeter} = a + b + c\]

Where \(a, b, c\) are the triangle's side lengths. Calculating the perimeter is straightforward when all side lengths are known.
In cases where the side lengths are determined from angles and other given measurements (like using the sine rule or an area-based formula), solving for the perimeter involves further calculations. Consider this scenario: if one side length is provided and using rules like the sine rule, you determine other sides, you can then sum these to find the perimeter. Understanding perimeter helps in applications such as constructing shapes and understanding proportions in similar triangles.