How to Find Solution of a Triangle? Complete Guide with Formulas, Examples, Short Notes, MCQs and Worksheet

Introduction

In geometry and trigonometry, a triangle is a three-sided polygon having three angles. In many mathematical problems, some sides and angles of a triangle are known while one or more sides or angles are unknown. The process of finding all the unknown sides and angles is called the solution of a triangle.

Knowing how to find solution of a triangle is an important skill in trigonometry. Different methods are used depending on the information given in the question.

The most commonly used methods involve:

• Angle Sum Property
• Pythagorean Theorem
• Trigonometric Ratios
• Sine Rule
• Cosine Rule

In this article, we will learn finding solution of a triangle step by step with simple examples and student-friendly notes.

What Is the Solution of a Triangle?

The solution of a triangle means finding all the unknown sides and angles when sufficient information about the triangle is given.

For example, suppose a triangle has:

• Two known angles
• One known side

We can use the angle sum property and trigonometric relationships to determine the remaining angle and sides.

A complete solution normally gives the values of:

• All three angles
• All three sides

Remember that the three interior angles of every triangle have a sum of 180°.

Important Parts of a Triangle

Consider a triangle ABC.

Its sides are:

• a = side opposite angle A
• b = side opposite angle B
• c = side opposite angle C

Its angles are:

• ∠A
• ∠B
• ∠C

The basic angle relationship is:

A + B + C = 180°

This relationship is useful for finding an unknown angle when the other two angles are known.

Conditions Required to Solve a Triangle

A triangle can be solved when enough information is given to determine its unknown parts.

Common cases include:

1. Three Sides Given — SSS

If all three sides are known, the triangle can be solved using the Cosine Rule to find the angles.

2. Two Sides and Included Angle — SAS

If two sides and the angle between them are known, the Cosine Rule can be used first to find the third side. The remaining angles can then be found using appropriate trigonometric relationships.

3. Two Angles and One Side — ASA or AAS

First find the third angle using:

A + B + C = 180°

Then the Sine Rule can be used to find the unknown sides.

4. Two Sides and an Opposite Angle — SSA

This case can sometimes have more than one possible triangle. Therefore, it is important to check the given information carefully before finding the solution.

5. Right-Angled Triangle

If one angle is 90°, trigonometric ratios and the Pythagorean Theorem can be used.

How to Find Solution of a Triangle

The general procedure for how to find solution of a triangle is:

Step 1: Draw and Label the Triangle

Draw the triangle and label the known sides and angles.

For example:

• Side a is opposite angle A.
• Side b is opposite angle B.
• Side c is opposite angle C.

Step 2: Identify the Given Information

Look carefully at the question and determine which sides and angles are known.

Ask yourself:

• Are three sides given?
• Are two sides and an angle given?
• Are two angles and one side given?
• Is the triangle right-angled?

Step 3: Find the Missing Angle

If two angles are known, use:

A + B + C = 180°

Therefore:

C = 180° − A − B

Step 4: Select the Appropriate Formula

Use the formula that matches the information given.

For right triangles, use trigonometric ratios or the Pythagorean Theorem.

For non-right triangles, the Sine Rule or Cosine Rule is often useful.

Step 5: Calculate the Unknown Parts

Substitute the known values into the appropriate formula and calculate the unknown side or angle.

Step 6: Check Your Answer

Finally, check that:

• All angles are positive.
• The three angles add up to 180°.
• The side lengths satisfy the triangle inequality.
• Your calculated values are reasonable.

Important Formulas for Finding Solution of a Triangle

1. Angle Sum Property

The sum of the three interior angles of a triangle is:

A + B + C = 180°

This is one of the most important formulas used in solving triangles.

2. Pythagorean Theorem

For a right-angled triangle:

a² + b² = c²

where c is the hypotenuse.

For example, if the two perpendicular sides are 3 cm and 4 cm:

c² = 3² + 4²

c² = 9 + 16

c² = 25

Therefore:

c = 5 cm

3. Trigonometric Ratios

For a right-angled triangle, the three basic trigonometric ratios are:

sin θ = Opposite / Hypotenuse

cos θ = Adjacent / Hypotenuse

tan θ = Opposite / Adjacent

These ratios are useful when a side and an angle are known.

4. Sine Rule

For a general triangle:

a / sin A = b / sin B = c / sin C

The Sine Rule is particularly useful when corresponding side-angle information is available.

It can be rearranged as:

a = b sin A / sin B

and similarly for other sides.

5. Cosine Rule

The Cosine Rule is:

a² = b² + c² − 2bc cos A

Similarly:

b² = a² + c² − 2ac cos B

c² = a² + b² − 2ab cos C

The Cosine Rule is especially useful when three sides are known or when two sides and the included angle are known.

Example 1: Finding an Unknown Angle

Suppose:

A = 60°

B = 70°

Find C.

Using:

A + B + C = 180°

60° + 70° + C = 180°

130° + C = 180°

Therefore:

C = 50°

So, the three angles are:

60°, 70°, 50°

Example 2: Solving a Right Triangle

Suppose a right-angled triangle has perpendicular sides of 6 cm and 8 cm. Find the hypotenuse.

Using the Pythagorean Theorem:

c² = a² + b²

c² = 6² + 8²

c² = 36 + 64

c² = 100

Therefore:

c = 10 cm

Example 3: Using the Sine Rule

Suppose:

A = 30°

B = 45°

a = 6 cm

Find b.

Using the Sine Rule:

a / sin A = b / sin B

Substitute the values:

6 / sin 30° = b / sin 45°

Since:

sin 30° = 0.5

sin 45° ≈ 0.7071

Therefore:

6 / 0.5 = b / 0.7071

12 = b / 0.7071

b ≈ 8.49 cm

Therefore:

b ≈ 8.49 cm

Example 4: Using the Cosine Rule

Suppose two sides of a triangle are:

b = 5 cm

c = 7 cm

and the included angle is:

A = 60°

Find side a.

Using the Cosine Rule:

a² = b² + c² − 2bc cos A

Substitute:

a² = 5² + 7² − 2(5)(7)cos 60°

a² = 25 + 49 − 70(0.5)

a² = 74 − 35

a² = 39

Therefore:

a = √39 ≈ 6.24 cm

Why Is Finding the Solution of a Triangle Important?

Finding solution of a triangle is useful in many areas of mathematics and practical applications.

It is used in:

• Trigonometry
• Geometry
• Surveying
• Construction
• Engineering
• Navigation
• Architecture
• Physics

Students should therefore understand not only the formulas but also when to use each formula.

Common Mistakes While Solving Triangles

Students often make the following mistakes:

Mistake 1: Forgetting the Angle Sum

The angles of a triangle must add up to 180°.

Mistake 2: Using the Wrong Side-Angle Pair

In the Sine Rule, each side must correspond to its opposite angle.

For example:

a ↔ A

b ↔ B

c ↔ C

Mistake 3: Confusing Hypotenuse and Other Sides

In a right-angled triangle, the hypotenuse is always opposite the 90° angle.

Mistake 4: Calculator in the Wrong Mode

When calculating trigonometric functions involving degrees, make sure the calculator is set to degree mode.

Mistake 5: Not Checking the Answer

Always check whether the calculated angles and sides make sense.

Short Notes: Solution of a Triangle

Definition

The process of finding all unknown sides and angles of a triangle from the given information is called the solution of a triangle.

Angle Sum

A + B + C = 180°

Pythagorean Theorem

For a right triangle:

a² + b² = c²

Trigonometric Ratios

sin θ = Opposite / Hypotenuse

cos θ = Adjacent / Hypotenuse

tan θ = Opposite / Adjacent

Sine Rule

a/sin A = b/sin B = c/sin C

Cosine Rule

a² = b² + c² − 2bc cos A

Common Cases

• SSS: Three sides known
• SAS: Two sides and included angle known
• ASA: Two angles and included side known
• AAS: Two angles and a non-included side known
• SSA: Two sides and an opposite angle known

Basic Steps

1. Draw the triangle.
2. Label the known values.
3. Identify the type of problem.
4. Select the appropriate formula.
5. Calculate the unknown values.
6. Check the answer.

MCQs on Solution of a Triangle

MCQ 1

The sum of the interior angles of a triangle is:

A) 90°

B) 180°

C) 270°

D) 360°

Answer: B) 180°

MCQ 2

Which theorem is used to solve for the hypotenuse of a right triangle when the other two sides are known?

A) Sine Rule

B) Cosine Rule

C) Pythagorean Theorem

D) Angle Sum Rule

Answer: C) Pythagorean Theorem

MCQ 3

The correct Sine Rule is:

A) a sin A = b sin B

B) a/sin A = b/sin B = c/sin C

C) a² = b² + c²

D) a + b + c = 180°

Answer: B

MCQ 4

The side opposite the 90° angle is called the:

A) Base

B) Adjacent side

C) Hypotenuse

D) Opposite side

Answer: C) Hypotenuse

MCQ 5

If two angles of a triangle are 50° and 60°, the third angle is:

A) 60°

B) 70°

C) 80°

D) 90°

Answer: B) 70°

MCQ 6

Which trigonometric ratio is equal to Opposite/Hypotenuse?

A) cos θ

B) tan θ

C) sin θ

D) sec θ

Answer: C) sin θ

MCQ 7

Which rule is particularly useful when three sides of a triangle are known?

A) Cosine Rule

B) Sine Rule

C) Tangent Rule

D) Angle Sum Rule only

Answer: A) Cosine Rule

MCQ 8

In the notation of a triangle, side a is opposite:

A) Angle B

B) Angle C

C) Angle A

D) 90°

Answer: C) Angle A

MCQ 9

If A = 90°, B = 40°, then C is:

A) 40°

B) 50°

C) 60°

D) 70°

Answer: B) 50°

MCQ 10

The Cosine Rule contains which trigonometric function?

A) sin

B) tan

C) cos

D) cot

Answer: C) cos

Worksheet / Assignment

Part A: Fill in the Blanks

1. The sum of the angles of a triangle is ________.
2. The side opposite the right angle is called the ________.
3. The Pythagorean Theorem is ________.
4. In a triangle, side a is opposite angle ________.
5. The Sine Rule contains the ratios of sides to the ________ of their opposite angles.

Part B: Find the Missing Angle

1. A = 40°, B = 70°. Find C.
2. A = 65°, B = 55°. Find C.
3. A = 90°, B = 35°. Find C.
4. A = 75°, B = 45°. Find C.
5. A = 52°, B = 68°. Find C.

Part C: Use the Pythagorean Theorem

1. A right triangle has sides 3 cm and 4 cm. Find the hypotenuse.
2. A right triangle has perpendicular sides 5 cm and 12 cm. Find the hypotenuse.
3. The hypotenuse of a right triangle is 13 cm and one side is 5 cm. Find the other side.

Part D: Use Trigonometric Ratios

1. Write the formula for sin θ.
2. Write the formula for cos θ.
3. Write the formula for tan θ.
4. A right triangle has an angle of 30° and a hypotenuse of 10 cm. Find the side opposite the angle using the Sine Ratio.
5. A right triangle has an angle of 60° and an adjacent side of 5 cm. Find the hypotenuse using the Cosine Ratio.

Part E: Sine Rule and Cosine Rule

1. State the Sine Rule.
2. State the Cosine Rule.
3. Two sides of a triangle are 5 cm and 7 cm and the included angle is 60°. Find the third side.
4. In a triangle, A = 30°, B = 45° and a = 6 cm. Find b using the Sine Rule.
5. Explain when the Cosine Rule can be used to solve a triangle.

Part F: Challenge Questions

1. Explain the difference between SSS, SAS, ASA and AAS.
2. A triangle has angles 45° and 65°. Find the third angle and explain which property you used.
3. Explain how you would decide whether to use the Sine Rule or Cosine Rule.
4. Write the complete steps you would follow for finding solution of a triangle.
5. Create your own triangle problem and solve it completely.

Answers to Worksheet

Part A

1. 180°
2. Hypotenuse
3. a² + b² = c²
4. A
5. Sine

Part B

1. 70°
2. 60°
3. 55°
4. 60°
5. 60°

Part C

1. 5 cm
2. 13 cm
3. 12 cm

Part D

1. sin θ = Opposite/Hypotenuse
2. cos θ = Adjacent/Hypotenuse
3. tan θ = Opposite/Adjacent
4. 5 cm
5. 10 cm

Part E

1. a/sin A = b/sin B = c/sin C
2. a² = b² + c² − 2bc cos A
3. Approximately 6.24 cm
4. Approximately 8.49 cm
5. It is useful when three sides are known or when two sides and their included angle are known.

Final Summary

The solution of a triangle means finding its unknown sides and angles from the information provided. To solve a triangle correctly, first identify what is known and then select the appropriate mathematical relationship.

The most important tools are:

• Angle Sum Property: A + B + C = 180°
• Pythagorean Theorem: a² + b² = c²
• Trigonometric Ratios: sine, cosine and tangent
• Sine Rule: a/sin A = b/sin B = c/sin C
• Cosine Rule: a² = b² + c² − 2bc cos A

Students can use these methods to solve a wide range of triangle problems in geometry and trigonometry.

For more online math academy notes, recorded lectures, assessments, lesson plans and other mathematics learning material, visit Nisar Math Academy at nisarmathacademy.com. Free resources are also available for students, while full-course content is available through website membership.

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