Hard Level Worksheet Questions

Triangles are fundamental geometric figures with essential properties including angle relationships, side constraints, and the Pythagorean theorem. Mastering these concepts is crucial for advanced geometry, trigonometry, and real-world applications in engineering and design.

Let's explore advanced triangle properties and relationships.

1. The three angles of a triangle are in the ratio 1 : 2 : 3. Find the smallest angle.

Step 1: Let angles be x, 2x, 3x

Step 2: Sum of angles: x + 2x + 3x = 180°360°90°270°

Step 3: x = 30°25°35°20°

Perfect! The smallest angle is 30° (ratio 1 : 2 : 3 gives 30° : 60° : 90°).

2. If one acute angle of a right triangle is twice the other, find both angles.

Step 1: Let smaller acute angle = x, larger =

Step 2: In right triangle: x + 2x + ° = 180°

Step 3: x = 30°25°35°20°

Where 2x = °

Therefore, angles are 30° and 60°50°70°40°

Excellent! The acute angles are 30° and 60° in this special right triangle.

3. A triangle has two angles measuring 65° and 45°. Find the third angle.

Let the angle be x.

Sum of all angles of a triangle = °

Step 1: Thesefore, x = 70°75°65°80°

Correct! Sum of angles in triangle = 180°, so third angle = 70°.

4. The sides of a triangle are 5 cm, 12 cm, and 13 cm. Identify the type of triangle.

Step 1: Check Pythagorean theorem: 5² + 12² = 25 + 144169130120 = 169150180200

Step 2: 13² = 169150180200

Step 3: Since 5² + 12² = 13², it's a Right-angled triangleAcute triangleObtuse triangleEquilateral triangle

Perfect! This is a right-angled triangle (5-12-13 Pythagorean triple).

5. State the triangle inequality property.

Sum of any two sides > third sideAll sides must be equalAngles must sum to 180°Largest angle opposite longest side

Excellent! Triangle inequality: sum of any two sides must be greater than the third side.

6. Find the length of the side of a square whose diagonal is 8√2 cm.

Step 1: In square, diagonal = side × √2side × 2

Step 2: side × √2 = 8√2

Step 3: side = 8 cm6 cm10 cm12 cm

Great! Side of square = diagonal ÷ √2 = 8√2 ÷ √2 = 8 cm.

7. The sides of a triangle are 8 cm, 15 cm, and 17 cm. Is it a right triangle?

Step 1: Check: 8² + 15² = 289290280270

Step 2: 17² = 289290280270

Step 3: Since 8² + 15² =≠ 17², answer is YesNo

Perfect! This is another Pythagorean triple: 8-15-17.

8. Find the value of x: angles are (x + 10)°, (2x – 5)°, and (3x – 15)°.

Step 1: Sum of angles = °

(x + 10) + (2x - 5) + (3x - 15) = 180°

Step 2: x = 31.67°30°32°35°

Excellent! Solving the equation gives x ≈ 31.67°.

Drag each property to its correct triangle classification:

Part B: Short Answer Questions (2 Marks Each)

1. The sides of a triangle are in the ratio 3 : 4 : 5, and its perimeter is 72 cm. Find the sides.

Step 1: Set up equation

Let sides be 3x, 4x, 5x

Perimeter: 3x + 4x + 5x = cm

Step 2: Solve for x

x = 6578

Step 3: Find the sides

Sides are: 3×6 = 18 cm15 cm21 cm24 cm, 4×6 = 24 cm20 cm28 cm32 cm, 5×6 = 30 cm25 cm35 cm40 cm

Perfect! The sides are 18 cm, 24 cm, and 30 cm.

2. Check whether a triangle with sides 9 cm, 40 cm, and 41 cm is right-angled triangle.

Step 1: Apply Pythagorean theorem

Check if: 9² + 40²41² = 41²40²

9² + 40² = 1681168016901700

Step 2: Calculate 41²

41² = 1681168016901700

Step 3: Compare results

Since 9² + 40² =≠ 41², it is a right-angledacuteobtusescalene triangle

Excellent! This is a right-angled triangle.

3. Two sides of a triangle are 8 cm and 6 cm. Between what limits must the third side lie?

Step 1: Apply triangle inequality

Lower limit: |8 - 6| = 2 cm1 cm3 cm4 cm

Step 2: Find upper limit

Upper limit: 8 + 6 = 14 cm12 cm16 cm10 cm

Step 3: State the range

Third side must be between 2 cm1 cm3 cm0 cm and 14 cm15 cm13 cm16 cm

Perfect! Triangle inequality gives range: 2 cm < third side < 14 cm.

4. The hypotenuse of a right triangle is 26 cm, and one side is 10 cm. Find the other side.

Step 1: Apply Pythagorean theorem

10² + x² = 26²

Step 2: Solve for other side

x² = 576600550580

Step 3: Find the side length

x = 24 cm22 cm26 cm20 cm

Excellent! Using Pythagorean theorem: other side = 24 cm.

Part C: Long Answer Questions (4 Marks Each)

1. The perimeter of a triangle is 60 cm. Its sides are in the ratio 5 : 6 : 7. Find the sides and check if it's right-angled.

Step 1: Find the sides

Let sides be 5x, 6x, 7x

Perimeter: 5x + 6x + 7x = [60] cm

x = 3.33343.5

Step 2: Calculate actual sides

Sides are: 5×3.33 = 16.67 cm15 cm18 cm20 cm, 6×3.33 = 20 cm18 cm22 cm24 cm, 7×3.33 = 23.33 cm21 cm25 cm28 cm

Step 3: Check if right-angled

Check: (16.67)² + (20)² = 677.89678680675

(23.33)² = 544.29545550540

Step 4: Conclusion

Since 677.89 ≠= 544.29, it's not right-angledright-angledacuteobtuse

Excellent! Sides are 16.67, 20, 23.33 cm, and it's not a right triangle.

2. In a right triangle, legs are in ratio 3:4, and hypotenuse is 25 cm. Find the other two sides.

Step 1: Set up equation

Let legs be 3x and 4x

Using Pythagorean theorem: (3x)² + (4x)² = 25²

Step 2: Solve equation

x² =

x² = , so x = 5463

Step 3: Find the legs

First leg = 3×5 = 15 cm12 cm18 cm20 cm

Second leg = 4×5 = 20 cm16 cm24 cm25 cm

Perfect! The legs are 15 cm and 20 cm (3-4-5 triangle scaled by 5).

3. A triangle has sides 13 cm, 20 cm, and 21 cm. Check if it's right triangle and verify triangle inequality.

Step 1: Check Pythagorean theorem

13² + 20² = 569570560580

21² = 441440450430

Step 2: Compare results

Since 569 ≠= 441, it's not a right trianglea right triangleimpossibleequilateral

Step 3: Verify triangle inequality

13 + 20 = >=< 21

13 + 21 = >=< 20

20 + 21 = >=< 13

All conditions satisfied, so triangle is possibleimpossibleright-angledequilateral

Excellent! Not a right triangle, but satisfies triangle inequality.

4. A 15 m ladder leans against a wall. Foot is 9 m from wall. Find height and check if triangle is right-angled.

Step 1: Apply Pythagorean theorem

height² + 9² = 15²

Step 2: Solve for height

height² = 144140150135

height = 12 m11 m13 m10 m

Step 3: Check if right-angled

Since ladder, ground, and wall meet at 90°, triangle is right-angledacuteobtusescalene

Perfect! Height = 12 m, forming a right triangle (9-12-15 Pythagorean triple).

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. In any triangle, the sum of the three angles is:

(a) 90° (b) 120° (c) 180° (d) 360°

Correct! The sum of interior angles in any triangle is always 180°.

2. A triangle with sides 6 cm, 8 cm, and 10 cm is:

(a) Equilateral (b) Acute-angled (c) Right-angled (d) Obtuse-angled

Correct! 6² + 8² = 36 + 64 = 100 = 10², so it's a right-angled triangle.

3. The triangle inequality property states that:

(a) The sum of all three sides > 180° (b) The sum of any two sides > the third side (c) The sum of angles > the largest angle (d) The sum of all sides = 180°

Correct! Triangle inequality: sum of any two sides must be greater than the third side.

4. In a right triangle, the side opposite the right angle is called:

(a) Base (b) Altitude (c) Hypotenuse (d) Median

Correct! The hypotenuse is the side opposite the right angle and is the longest side.

5. A triangle whose all angles are less than 90° is called:

(a) Obtuse-angled (b) Right-angled (c) Acute-angled (d) Equilateral

Correct! An acute-angled triangle has all three angles less than 90°.

🎉 Outstanding! You've Mastered Hard Level Triangle Properties! Here's what you accomplished:

✓ Advanced Angle Relationships: Ratio problems, angle sum applications, and special triangles

✓ Pythagorean Theorem Mastery: Identifying right triangles, calculating unknown sides

✓ Triangle Inequality Understanding: Determining possible side lengths and triangle existence

✓ Triangle Classifications: Acute, right, obtuse, equilateral, isosceles, and scalene triangles

✓ Real-World Applications: Ladder problems, construction scenarios, and practical geometry

✓ Complex Problem Solving: Multi-step problems involving ratios, perimeters, and calculations

✓ Pythagorean Triples: Recognizing and working with common right triangle ratios

✓ Geometric Reasoning: Logical analysis of triangle properties and relationships

Your expertise in triangle properties prepares you for advanced trigonometry, coordinate geometry, and engineering applications!