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AFCAT Solution of Triangles

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This page covers AFCAT Solution of Triangles with complete concept notes, 33 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Solution of Triangles— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

Angle Sum

A + B + C = 180 degrees. Always. 2

Sides are opposite their angles

side 'a' is opposite angle A, side 'b' is opposite angle B, side 'c' is opposite angle C. 3. Larger side is always opposite the larger angle.

2.Formula Block

Memorise — at least one formula appears in every paper

a / sin A = b / sin B = c / sin C = 2R

(R = circumradius of the triangle)

a² = b² + c² - 2bc·cos A

b² = a² + c² - 2ac·cos B

c² = a² + b² - 2ab·cos C

tan((B - C)/2) = ((b - c)/(b + c)) · cot(A/2)

Area = (1/2)·b·c·sin A = (1/2)·a·c·sin B = (1/2)·a·b·sin C

Also: Area = sqrt(s(s-a)(s-b)(s-c)) [Heron's Formula], where s = (a+b+c)/2

Circumradius: R = a / (2·sin A) = abc / (4·Area)

Inradius: r = Area / s

Sine Rule:

Cosine Rule:

Tangent Rule (Napier's Analogy):

Area of Triangle:

3.Exam Patterns

What examiners ask — read before attempting PYQs

NDA frequently asks: - Find unknown side or angle using Sine Rule or Cosine Rule - Find area of triangle given two sides and included angle - Find circumradius R or inradius r - Problems where angle sum property is needed first SHORTCUT / TRICK Trick 1 —

4.When to Use

Quickly decide which method to apply in the exam

which rule: * Two angles + one side given → Use Sine Rule directly. It is faster. * Two sides + included angle given → Use Cosine Rule to find the third side. * All three sides given → Use Cosine Rule to find any angle. Trick 2 — Quick R formula: In an equilateral triangle with side a: R = a / sqrt(3) and r = a / (2·sqrt(3)). Memorize: R = 2r for equilateral triangle is WRONG.

Correct ratio is R = 2r.

5.Worked Example

Solve this step-by-step before moving on

1

Step 1

Use Cosine Rule for angle A. cos A = (b² + c² - a²) / (2·b·c)

2

Step 2

Substitute values. cos A = (64 + 81 - 49) / (2 × 8 × 9) cos A = 96 / 144 cos A = 2/3

3

Step 3

Find A. A = cos⁻¹(2/3) ≈ 48.19 degrees Answer: Angle A ≈ 48.19 degrees

6.Exam Traps

Common mistakes students make — avoid these

Students confuse which formula to use. Remember: Sine Rule needs a known angle opposite a known side. If you only have sides and no angle, start with Cosine Rule.

Also, never forget that the sum A + B + C = 180 — once two angles are found, the third is automatic.

Key Points to Remember

Sine Rule: a/sin A = b/sin B = c/sin C = 2R, where R is circumradius
Cosine Rule: a² = b² + c² - 2bc·cos A (and cyclic forms for b and c)
Use Sine Rule when two angles and one side are known
Use Cosine Rule when all three sides, or two sides and included angle, are known
Area of triangle = (1/2)·b·c·sin A — use when two sides and included angle are given
Inradius r = Area / s, where s = (a+b+c)/2 is the semi-perimeter
Circumradius R = abc / (4·Area) — important formula for MCQs
Angle sum property: A + B + C = 180 degrees — always use this to find the third angle

Exam-Specific Tips

Sine Rule connects all three sides and angles: a/sin A = b/sin B = c/sin C = 2R
Circumradius R = a / (2·sin A) — direct formula used in MCQs
Inradius r = Area / s where s = semi-perimeter = (a+b+c)/2
For an equilateral triangle with side a: R = a/sqrt(3) and r = a/(2·sqrt(3)), giving R = 2r
Heron's Formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2
Cosine Rule rearranged: cos A = (b² + c² - a²) / (2bc) — used to find angles from sides
Napier's Analogy: tan((B-C)/2) = ((b-c)/(b+c))·cot(A/2) — used in advanced problems
In a right-angled triangle with hypotenuse c and right angle C: c = 2R, so the hypotenuse equals the diameter of the circumcircle

Practice MCQs

Solution of Triangles — Practice Questions

33graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 33

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Practice 1easy

In triangle ABC, if a = 13, b = 14, c = 15, then sin(A/2) is equal to:

Practice 2easy

In triangle ABC, if a = 4, b = 5, c = 6, then tan(C/2) is equal to:

Practice 3easy

In triangle ABC, a = 5, b = 6, c = 7. Using the cosine rule, find cos C and then determine whether angle C is acute or obtuse.

Practice 4easy

In triangle ABC, if a = 13, b = 14, c = 15, then the value of sin(A/2) is:

Practice 5easy

In triangle ABC, if a/sin A = b/sin B = c/sin C = k, then k is equal to:

Practice 6easy

In triangle ABC, if cos A + cos B + cos C = 1 + 4sin(A/2)sin(B/2)sin(C/2), then the triangle is:

Practice 7easy

In triangle ABC, a = 4, b = 5, c = 6. The value of cos(B/2) is:

Practice 8easy

In triangle ABC, if sin A = 3/5 and sin B = 5/13, then sin C is equal to:

Practice 9easy

In triangle ABC, if a/sin A = b/sin B = c/sin C = 2R (where R is the circumradius), and a = 6, b = 8, then the value of sin A : sin B is:

Practice 10easy

In triangle ABC, if a = 13, b = 14, c = 15, find cos B.

Practice 11easy

In triangle ABC, a = 7, b = 8, c = 9. Find sin A using the sine rule and the area formula.

Practice 12easy

In triangle ABC, if a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius, and a = 10, A = 60°, find R.

Practice 13easy

In triangle ABC, if A = 45°, B = 60°, and a = 2√2, find side b using the sine rule.

Practice 14easy

In triangle ABC, a = 7, b = 8, c = 9. The value of cos(B/2) is:

Practice 15easy

In triangle ABC, if a² + b² - c² = ab, then angle C is equal to:

Practice 16medium

In triangle ABC, if a² + b² - c² = ab, then angle C equals:

Practice 17medium

In triangle ABC, if a = 13, b = 14, c = 15, then the value of sin(A/2) is:

Practice 18medium

In triangle ABC, a = 7, b = 8, c = 9. The value of cos(B/2) is:

Practice 19medium

In triangle ABC, if a = 5, b = 6, c = 7, then tan(A/2) is equal to:

Practice 20medium

In triangle ABC, if sin A = 3/5 and sin B = 5/13, then sin C is equal to:

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60-Second Revision — Solution of Triangles

Formula: Sine Rule — a/sin A = b/sin B = c/sin C = 2R. Use when angle-side pair is known.
Formula: Cosine Rule — a² = b² + c² - 2bc·cos A. Use when all sides or two sides + included angle are given.
Formula: Area = (1/2)·b·c·sin A and also Area = sqrt(s(s-a)(s-b)(s-c)).
Remember: Circumradius R = abc/(4·Area) and Inradius r = Area/s.
Trap: Do not use Sine Rule if no angle-side pair is given — switch to Cosine Rule instead.
Remember: Once two angles are found, always get the third using A + B + C = 180.
Trick: In right triangle, hypotenuse = diameter of circumcircle, so R = c/2.

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