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NDA Triangles — Area & Properties

Study Material — 24 PYQs (2018–2022) · Concept Notes · Shortcuts

NDA Triangles — Area & Properties is a frequently tested subtopic — 24 previous year questions from 2018–2022 papers are included below with concept notes, key rules and shortcut tricks.

24 PYQs

2018–2022

19 Practice

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10 Key Points

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Previous Year Questions

NDA Triangles — Area & Properties — Past Exam Questions

24 questions from actual NDA papers · all shown free · click option to reveal solution

Exam Q 12022Previous Year Pattern

An equilateral triangle has a side length of 10 cm. What is the length of its altitude (height)?

Exam Q 22022Previous Year Pattern

The area of a triangle is 60 cm² and its base is 15 cm. What is the height of the triangle?

Exam Q 32022Previous Year Pattern

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its area using Heron's formula?

Exam Q 42022Previous Year Pattern

The area of a triangle is 24 cm² and its base is 6 cm. What is the height of the triangle?

Exam Q 52022Previous Year Pattern

An equilateral triangle has a side length of 10 cm. What is its perimeter?

Exam Q 62018Previous Year Pattern

The base of a triangle is 14 cm and its height is 10 cm. What is the area of the triangle?

Exam Q 72022Previous Year Pattern

A triangle has a base of 12 cm and a height of 8 cm. What is its area?

Exam Q 82022Previous Year Pattern

A right-angled triangle has legs of length 9 cm and 12 cm. What is the length of its hypotenuse?

Exam Q 92022Previous Year Pattern

A right-angled triangle has legs of length 6 cm and 8 cm. What is the length of its hypotenuse?

Exam Q 102022Previous Year Pattern

A triangle has a base of 16 cm and a height of 12 cm. What is its area?

Exam Q 112022Previous Year Pattern

Triangle ABC has an area of 60 cm². Point D is on side AB such that AD:DB = 3:2. Point E is on side AC such that AE:EC = 3:2. What is the area of triangle ADE?

Exam Q 122022Previous Year Pattern

A right-angled triangle has legs of 9 cm and 12 cm. A circle is inscribed in this triangle. What is the radius of the inscribed circle?

Exam Q 132022Previous Year Pattern

A triangle has vertices at A(0, 0), B(8, 0), and C(4, 6). What is the area of the triangle?

Exam Q 142018Previous Year Pattern

The sides of a triangle are 13 cm, 14 cm, and 15 cm. What is the area of the triangle (in cm²)?

Exam Q 152022Previous Year Pattern

A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area?

Exam Q 162022Previous Year Pattern

In triangle ABC, the median from A to side BC has length 10 cm. If the median from B to side AC has length 12 cm, and these two medians intersect at the centroid G, what is the distance AG?

Exam Q 172022Previous Year Pattern

In triangle ABC, AB = 10 cm, AC = 15 cm, and angle BAC = 60°. The area of triangle ABC is k√3 cm². Find k.

Exam Q 182022Previous Year Pattern

A triangle has vertices at A(0, 0), B(8, 0), and C(3, 6). A line parallel to AB passes through the centroid of triangle ABC and intersects AC at point P and BC at point Q. Find the length of PQ (in units).

Exam Q 192022Previous Year Pattern

A triangle has sides 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. What is the length of this perpendicular (in cm)?

Exam Q 202022Previous Year Pattern

A right-angled triangle has legs of 9 cm and 12 cm. A circle is inscribed in this triangle. Find the radius of the inscribed circle (in cm).

Exam Q 212022Previous Year Pattern

In triangle ABC, the angle bisector from vertex A meets side BC at point D. If AB = 18 cm, AC = 24 cm, and BD = 9 cm, what is the length of DC (in cm)?

Exam Q 222018Previous Year Pattern

In triangle ABC, the altitudes from vertices A, B, and C are 12 cm, 15 cm, and 20 cm respectively. If the area of the triangle is 60 cm², find the length of the side opposite to vertex A (i.e., side BC).

Exam Q 232022Previous Year Pattern

A triangle has sides 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. Find the length of this perpendicular (in cm).

Exam Q 242022Previous Year Pattern

In triangle PQR, the sides are PQ = 7 cm, QR = 8 cm, and PR = 9 cm. The altitude from Q to side PR has length h. Find h² (in cm²).

Concept Notes

Triangles — Area & Properties— Rules & Concept

Core ConceptRead this first — the foundation of the topic

Angle Sum Property

All three angles add up to 180°. 2

Exterior Angle Property

An exterior angle equals the sum of the two non-adjacent interior angles. 3

Triangle Inequality

Sum of any two sides is always greater than the third side. 4. The longest side is always opposite the largest angle. 5. In an equilateral triangle: all sides are equal, all angles are 60°. 6. In an isosceles triangle: two sides are equal, and the angles opposite those sides are equal. 7.

In a right-angled triangle: the square of the hypotenuse equals the sum of squares of the other two sides (Pythagoras theorem).

Formula BlockMemorise — at least one formula appears in every paper

• Basic Area = (1/2) × Base × Height

• Heron's Formula (when all 3 sides are known): Area = sqrt(s × (s-a) × (s-b) × (s-c)), where s = (a+b+c)/2

• Equilateral Triangle Area = (sqrt(3)/4) × a², where a = side

• Equilateral Triangle Height = (sqrt(3)/2) × a

• Right Triangle Area = (1/2) × base × perpendicular

• Area using two sides and included angle = (1/2) × a × b × sin(C)

• Inradius (r) = Area / s, where s is semi-perimeter

• Circumradius (R) = (a × b × c) / (4 × Area)

Exam PatternsWhat examiners ask — read before attempting PYQs

Trick 2 — Equilateral Triangle Fast Formula

For side = a, Area = 1.732/4 × a². Use 1.732 as value of sqrt(3). For a = 6, Area = 1.732/4 × 36 = 15.59 sq units

Trick 3 — Median Divides Triangle into Equal Areas

A median splits a triangle into two smaller triangles of exactly equal area. So if a triangle has area 48, each half = 24. The centroid divides the triangle into 6 equal smaller triangles.

Worked ExampleSolve this step-by-step before moving on

1

Step 1

Check if it is a right triangle. 5² + 12² = 25 + 144 = 169 = 13². Yes, it is a right triangle.

2

Step 2

Use Area = (1/2) × base × height = (1/2) × 5 × 12 = 30 sq cm. Answer: 30 sq cm. No need for Heron's formula here — smart recognition saves time. WORKED EXAMPLE 2 Q: An equilateral triangle has a perimeter of 36 cm. Find its area.

1

Step 1

Side = 36/3 = 12 cm.

2

Step 2

Area = (sqrt(3)/4) × 12² = (1.732/4) × 144 = 0.433 × 144 = 62.35 sq cm. Answer: 36√3 sq cm (exact) or approximately 62.35 sq cm.

Exam TrapsCommon mistakes students make — avoid these

— THE #1 TRAP Students confuse Height with Side in equilateral triangles. The HEIGHT of an equilateral triangle is (sqrt(3)/2) × a, NOT the side itself. Many students use the side as the height in the area formula and get the wrong answer.

Always remember: base × height, and height must be the perpendicular height, not a slant side.

Key Points to Remember

Sum of all angles in any triangle = 180°
Area of triangle = (1/2) × Base × Height — most used formula in SSC
Equilateral triangle area = (√3/4) × a², height = (√3/2) × a
Heron's Formula: Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
For 5-12-13 and 3-4-5 triangles: it is a right triangle — use (1/2)×b×h directly
Two triangles with same height: ratio of areas = ratio of their bases
A median divides a triangle into 2 equal area triangles; centroid creates 6 equal triangles
Inradius r = Area / Semi-perimeter; Circumradius R = (a×b×c) / (4×Area)
Exterior angle of a triangle = sum of the two opposite interior angles
Longest side is always opposite the largest angle in any triangle

Exam-Specific Tips

For an equilateral triangle with side 'a': Area = (√3/4)a², Height = (√3/2)a, Perimeter = 3a
Pythagorean triplets commonly tested: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
The centroid divides each median in ratio 2:1 from vertex to midpoint of opposite side
Inradius formula: r = Area / s, where s = semi-perimeter = (a+b+c)/2
Circumradius formula: R = (abc) / (4 × Area) — often asked in advanced SSC CGL
A triangle with sides 5, 12, 13 has area = 30 sq units (direct recall saves time)
The area of a triangle formed by joining the midpoints of a triangle = 1/4 of original triangle area
For right-angled triangle: Circumradius R = Hypotenuse / 2

Practice MCQs

Triangles — Area & Properties — Practice Questions

19graded MCQs · easy to hard · full solution & trap analysis

All MCQs →

Practice 1easy

A right-angled triangle has legs of length 6 cm and 8 cm. What is the length of its hypotenuse?

Practice 2easy

A triangle has a base of 12 cm and a height of 8 cm. What is its area?

Practice 3easy

An equilateral triangle has a side length of 10 cm. What is its perimeter?

Practice 4easy

The area of a triangle is 54 cm² and its base is 12 cm. What is its height?

Practice 5easy

An isosceles triangle has two equal sides of 13 cm each and a base of 10 cm. What is its perimeter?

Practice 6easy

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its semi-perimeter?

Practice 7easy

A triangle has a base of 16 cm and a height of 12 cm. What is its area?

Practice 8easy

The area of a triangle is 60 cm² and its base is 15 cm. What is the height of the triangle?

Practice 9medium

In triangle ABC, the altitude from vertex A to side BC is 12 cm. If the area of triangle ABC is 90 cm², and a line parallel to BC intersects AB at point P and AC at point Q such that AP:PB = 2:1, then the area of triangle APQ is:

Practice 10medium

A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area?

Practice 11medium

A triangle has sides of length 13 cm, 14 cm, and 15 cm. Using Heron's formula, find its area.

Practice 12medium

The area of an equilateral triangle is 36√3 cm². Find the length of its side.

Practice 13medium

A triangle has vertices at A(0, 0), B(8, 0), and C(4, 6). Find the area of the triangle using the coordinate formula.

Practice 14medium

An equilateral triangle has a side length of 8 cm. What is the length of its altitude?

Practice 15hard

In triangle ABC, the medians from vertices A and B intersect at the centroid G. If the median from A has length 18 cm and the median from B has length 24 cm, and these medians are perpendicular to each other at G, find the area of triangle ABC (in cm²).

Practice 16hard

A triangle has vertices at A(0, 0), B(8, 0), and C(4, 6). A line parallel to AB passes through the centroid of the triangle and intersects sides AC and BC at points P and Q respectively. Find the length of PQ (in cm).

Practice 17hard

A triangle has sides 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side, meeting it at point P. Find the length of this perpendicular (in cm).

Practice 18hard

In triangle ABC, the sides are in the ratio 3:4:5. If the area of the triangle is 96 cm², find the length of the longest side (in cm).

Practice 19hard

A triangle has sides 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. Find the length of this perpendicular (in cm).

60-Second Revision — Triangles — Area & Properties

Formula: Area = (1/2) × Base × Height — always use perpendicular height, never slant side
Formula: Equilateral triangle area = (√3/4) × a²; height = (√3/2) × a — do NOT use side as height
Trap: In equilateral triangle, height ≠ side. Height = (√3/2) × a. This is the #1 mistake in exams
Shortcut: Same height triangles — area ratio = base ratio. No calculation needed
Remember: Pythagorean triplets (3-4-5, 5-12-13, 8-15-17) — spot right triangles instantly
Formula: Heron's Area = √(s(s-a)(s-b)(s-c)); compute s first, then substitute
Remember: Median splits triangle into 2 equal areas; centroid creates 6 equal smaller triangles

Circles — Area & Circumference→

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