Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

In this hard level, we'll explore advanced construction techniques, analyze triangle properties critically, and solve complex geometric problems.

Let's challenge our understanding with precise constructions and mathematical reasoning.

1. Define the SAS condition of triangle construction.

SAS stands for Side-Angle-SideSide-Side-Angle

In SAS, sides and the includedany angle between them are given

Perfect! The angle must be between the two given sides for unique construction.

2. What happens if the sum of two sides equals the third side?

If sum equals third side, the triangle cannot be constructedbecomes a line

This is because the two sides will lie onalong the third side forming a straight linedegenerate triangle

Excellent! Triangle inequality requires: sum of two sides > third side, not equal.

3. Write the condition for constructing a triangle when base, angle, and side are given.

This condition is SAS (Side-Angle-Side)ASA

The given angle must be the included angleopposite angle between the base and the side

Correct! The angle's position is crucial for unique construction.

4. Write one difference between SSS and SAS construction.

SSS uses three sidestwo sides, while SAS uses two sidesthree sides and one included angleone side

Great! The key difference is whether we use sides only or sides with an angle.

5. Which condition is used when hypotenuse and one side are given?

Answer: RHS (Right-Hypotenuse-Side)SAS

This applies only to right-angledacute triangles

Perfect! RHS is a special case for right triangles.

6. Write two examples of real-life applications of triangle construction.

Example 1: Bridge constructionBuilding

Example 2: Roof truss designArchitecture or NavigationLand surveying

Excellent! Triangles provide stability in structures and help in surveying.

7. Can a triangle be constructed with sides 2 cm, 3 cm, and 5 cm? Explain.

NoYes, because 2 + 3 = which is equal toless than 5

For valid triangle: sum must be greater thanequal to third side

Correct! This violates triangle inequality (2 + 3 = 5, not > 5).

8. What is the triangle inequality rule?

The sum of any two sidesthree sides must be greater thanequal to the thirdremaining side

Perfect! This must be verified for all three combinations of sides.

9. What is the sum of all angles of a triangle?

Sum of all angles = °

This is called the angle sum propertytriangle rule of triangles

Great! This fundamental property applies to all triangles.

10. Mention any two triangle construction conditions.

Condition 1: SSSSAS (all three sides known)

Condition 2: SASASA (two sides and included angle) or ASARHS or RHSSSS

Excellent! There are four main conditions: SSS, SAS, ASA, and RHS.

Drag each construction data to its correct condition:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Construct ΔABC with sides AB = 6 cm, BC = 7 cm, and AC = 8 cm.

Draw on your answer sheet with accurate measurements.

Verify triangle inequality:

6 + 7 = > 8

6 + 8 = > 7

7 + 8 = > 6

Construction condition:

Steps:

Draw base BC = cm (can use any side as base)

From B, draw arc with radius cm

From C, draw arc with radius cm

Mark intersection as point

Join AB and AC to complete triangleconstruction

Excellent! ΔABC is a scalene triangle with all sides different.

2. Construct ΔPQR with PQ = 5 cm, ∠P = 50°, and PR = 7 cm.

Construction condition:

The angle ∠P is includedopposite between sides PQ and PR

Steps:

Draw PQ = cm

At point P, construct angle ° using protractor

On the ray from P, cut PR = cm with compass

Join points and

Measure QR ≈ cm (approximately)

Perfect! Triangle constructed using SAS condition.

3. Construct a right-angled triangle ABC in which AB = 6 cm and AC = 10 cm.

Note: AC is the hypotenuse (longest side in right triangle)

Find third side using Pythagoras theorem:

AC² = AB² + BC²

² = ² + BC²

100 = 36 + BC² → BC² =

BC = cm

Construction condition:

Steps:

Draw AB = cm

At B, construct ° angle (perpendicular)

From A, draw arc with radius cm (hypotenuse)

Mark intersection on perpendicular as

Join AC to complete the right triangletriangle

Excellent! This is a 6-8-10 Pythagorean triplet (multiple of 3-4-5).

4. Construct ΔXYZ with XY = 5 cm, YZ = 6 cm, and ∠Y = 45°.

Construction condition:

∠Y is the includedopposite angle between XY and YZ

Steps:

Draw XY = cm

At Y, construct ∠Y = °

On ray, cut YZ = cm

Join and

Measure XZ to verify construction

Great! Triangle XYZ constructed successfully.

5. Construct ΔLMN given LM = 4 cm, MN = 6 cm, and ∠M = 75°.

Construction condition:

Steps:

Draw base LM = cm

At M, construct angle °

Cut MN = cm on the ray

Join L and

Perfect! ΔLMN is complete.

6. Draw a triangle with sides 3.5 cm, 5 cm, and 6 cm, and verify with a scale.

Verify inequality:

3.5 + 5 = > 6

Construction condition:

Draw base = cm (largest side)

Arc from left: radius cm or cm

Arc from right: the remaining measurement

Verification: Measure all sides with ruler to confirm accuracy

Excellent! Always verify your construction with measurements.

7. Construct ΔDEF where DE = 7 cm, ∠E = 60°, and EF = 5 cm.

Construction condition:

Draw DE = cm

At E, make angle °

Cut EF = cm

Join and

Great work! ΔDEF constructed.

8. Construct an isosceles triangle with equal sides 6 cm and included angle 50°.

Construction condition:

In isosceles triangle, two sidesall sides are equal

Draw first side = cm

At one end, construct angle °

On ray, cut second equal side = cm

Join endpoints to form isosceles triangletriangle

The two base anglesequal angles will be equal

Perfect! Isosceles triangle has two equal sides and two equal angles.

9. Construct a right triangle where the base is 8 cm and hypotenuse 10 cm.

Find perpendicular side:

10² = 8² + h² → h² = - =

h = cm

Construction condition:

Draw base = cm

Construct ° at one end

Arc from other end: radius = cm

Mark intersection and join

Excellent! Another 6-8-10 right triangle.

10. Construct ΔABC given that AB = 5 cm, AC = 6 cm, and ∠A = 45°.

Construction condition:

Draw AB = cm

At A, construct °

Cut AC = cm

Join and

Perfect! All short answer constructions complete.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Construct ΔABC where AB = 5 cm, AC = 7 cm, and ∠A = 60°.

(a) Write all steps clearly:

Construction condition:

Step 1: Draw AB = cm using ruler

Step 2: At point A, place protractor and mark °

Step 3: Draw ray from A through the ° mark

Step 4: Using compass, measure cm and cut on ray from A

Step 5: Mark this point as

Step 6: Join points and with ruler

Step 7: Label the triangle as ΔABCABC

(b) Measure and verify the third side:

Measure BC with ruler: approximately to cm

Verify angles:

Measure ∠B ≈ degrees

Measure ∠C ≈ degrees

Check: 60° + ∠B + ∠C ≈ °

Excellent! Triangle verified successfully with all properties checked.

2. Construct a right triangle ABC in which base = 9 cm and hypotenuse = 15 cm.

(a) Write the steps:

First, find the perpendicular side:

Using Pythagoras: ² = ² + p²

225 = 81 + p² → p² =

p = cm

Construction steps:

Step 1: Draw base BC = cm

Step 2: At B, construct ° angle using set square or protractor

Step 3: Draw perpendicular ray from B

Step 4: From C, draw arc with radius = cm (hypotenuse)

Step 5: Mark intersection with perpendicular as point

Step 6: Join and

Step 7: ΔABC is complete

(b) Identify which congruency rule is used:

Rule used:

This rule applies only to right-angledacute triangles

This is a 9-12-152-8-10 Pythagorean triplet (multiple of )

Perfect! RHS construction verified. This is a 3-4-5 scaled triangle.

3. Construct ΔXYZ where XY = 5 cm, YZ = 6 cm, and ∠Y = 45°.

(a) Describe construction steps:

Construction condition:

Step 1: Draw base XY = cm

Step 2: At Y, construct angle ° with protractor

Step 3: On the ray from Y, mark YZ = cm using compass

Step 4: Join and to complete triangle

(b) Find ∠Z and name the triangle:

Measure angles:

∠Y = ° (given)

Measure ∠X with protractor ≈ degrees

Calculate ∠Z = 180° - 45° - ∠X ≈ degrees

Triangle type: This is a scaleneisosceles triangle (all sides differentequal)

It is also an acuteobtuse triangle (all angles < 90°)

Excellent! Complete analysis of the triangle.

4. Construct a triangle with sides 6 cm, 8 cm, and 10 cm.

(a) Verify if it is right-angled using the Pythagoras theorem:

Check Pythagorean relationship:

Largest side (hypotenuse) = cm

Other two sides: cm and cm

Check: ² + ² = ²

36 + 64 =

100 = 100 ✓

Conclusion: , this is a right-angledacute triangle

The right angle is opposite to the hypotenusebase (10 cm side)

(b) Write the steps clearly:

Construction condition: (or can use )

Step 1: Draw base = cm (largest side/hypotenuse)

Step 2: From left end, draw arc with radius cm

Step 3: From right end, draw arc with radius cm

Step 4: Mark intersection point

Step 5: Join to complete triangle

Verification: Measure the angle opposite 10 cm side = °

Perfect! This is the famous 3-4-5 Pythagorean triplet scaled by 2.

5. Construct ΔPQR in which PQ = 5 cm, PR = 6 cm, and ∠Q = 50°.

Note: This is a challenging problem! Angle at Q is given, not at P.

(a) Explain the construction process step-by-step:

Analysis: This is NOT standard SAS because angle is at , not between PQ and PR

Construction approach:

Step 1: Draw PQ = cm

Step 2: At Q, construct angle ° using protractor

Step 3: Draw a ray from Q making 50° angle

Step 4: From P, draw arc with radius cm (PR length)

Step 5: This arc should intersect the ray from Q at point

Step 6: Join P and R

(b) Measure all sides and verify triangle properties:

Measurements:

PQ = cm (given)

PR = cm (given)

Measure QR ≈ cm (depends on construction)

Verify angles:

∠Q = ° (given)

Measure ∠P with protractor

Measure ∠R with protractor

Check: ∠P + ∠Q + ∠R = °

Verify triangle inequality:

PQ + PR > QR

PQ + QR > PR

PR + QR > PQ

Excellent! This complex construction requires understanding of geometric relationships.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. The base of a triangle is drawn:

(a) First (b) Second (c) Last (d) Any order

Correct! We always start by drawing the base (one side) first in triangle construction.

7. Which condition is not valid for constructing a triangle?

(a) SSS (b) SAS (c) ASA (d) SSA

Perfect! SSA (Side-Side-Angle) doesn't guarantee a unique triangle - it's ambiguous.

8. For RHS construction, what must be known?

(a) One right angle, hypotenuse, and one side

(b) Two sides only

(c) All sides

(d) Only base

Correct! RHS requires: Right angle (90°) + Hypotenuse + One side.

9. The instrument used to measure angles is:

(a) Compass (b) Divider (c) Protractor (d) Ruler

Excellent! Protractor is used to measure and construct angles in degrees.

10. The side opposite to 90° in a triangle is:

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

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

🏆 Outstanding Achievement! You've Mastered Advanced Triangle Construction!

Here's what you've conquered at the hard level:

• Deep Understanding of Construction Conditions:

  SSS (Side-Side-Side):

  • All three sides must be known
  • Triangle inequality must be verified first
  • No angles needed; sides determine everything
  • Most rigid condition (no flexibility)

  SAS (Side-Angle-Side):

  • Two sides and the INCLUDED angle (critical!)
  • Angle must be between the two given sides
  • Most commonly used in practical problems
  • Guarantees unique triangle

  ASA (Angle-Side-Angle):

  • Two angles and the side BETWEEN them
  • Third angle can be calculated (180° - sum)
  • Side must be included between angles
  • Commonly used in surveying

  RHS (Right-Hypotenuse-Side):

  • Special case for right triangles ONLY
  • Requires: 90° angle + Hypotenuse + One other side
  • Third side can be found using Pythagoras
  • Very useful in architecture and engineering

  SSA - NOT VALID:

  • Side-Side-Angle is ambiguous
  • Can produce 0, 1, or 2 different triangles
  • Never use SSA for unique construction

• Triangle Inequality - Mathematical Foundation:

  • Critical Rule: a + b > c (for all combinations)
  • Must verify THREE conditions:
    1. Side₁ + Side₂ > Side₃
    2. Side₁ + Side₃ > Side₂
    3. Side₂ + Side₃ > Side₁
  • If sum EQUALS third side → Forms a straight line (degenerate)
  • If sum LESS than third side → Cannot form triangle
  • Always check BEFORE attempting construction

• Advanced Verification Techniques:

  For All Triangles:

  • Measure all three sides with ruler
  • Measure all three angles with protractor
  • Verify: ∠A + ∠B + ∠C = 180°
  • Check triangle inequality holds

  For Right Triangles:

  • Verify one angle is exactly 90°
  • Check Pythagoras: a² + b² = c²
  • Hypotenuse is always the longest side
  • Common triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25

• Pythagorean Triplets - Perfect Right Triangles:

  • Basic triplets: 3-4-5, 5-12-13, 8-15-17
  • Scaled versions: 6-8-10 (2×3-4-5), 9-12-15 (3×3-4-5)
  • Formula check: c² = a² + b² (c = hypotenuse)
  • Examples:
    • 5² + 12² = 25 + 144 = 169 = 13²
    • 6² + 8² = 36 + 64 = 100 = 10²

• Complex Construction Scenarios:

  Non-standard cases:

  • When angle is NOT included (like ∠Q given with PQ and PR)
  • Requires arc-intersection method
  • May need trial-and-error approach
  • Must verify final result carefully

  Isosceles triangles:

  • Two equal sides create two equal angles
  • Equal angles are opposite equal sides
  • Can be constructed using SAS with equal sides

  Equilateral triangles:

  • All sides equal (can use any as base)
  • All angles = 60°
  • Highly symmetric construction

• Real-Life Applications:

  1. Civil Engineering: Bridge trusses, roof structures
  2. Navigation: Triangulation for GPS, ship positioning
  3. Architecture: Building frameworks, stable structures
  4. Land Surveying: Measuring inaccessible distances
  5. Computer Graphics: 3D modeling, mesh generation
  6. Robotics: Arm positioning, movement calculations

• Professional Construction Tips:

  • Use sharp HB pencil for construction lines
  • Keep construction lines light initially
  • Draw final triangle with darker lines
  • Label all vertices clearly with capital letters
  • Show all arcs used in construction
  • Mark all given measurements on diagram
  • Use set squares for perfect right angles
  • Verify measurements after construction

• Common Advanced Mistakes to Avoid:

  • Assuming SSA is valid (it's NOT!)
  • Forgetting to verify triangle inequality
  • Using non-included angle in SAS
  • Not checking Pythagoras for right triangles
  • Inaccurate angle measurement (±1° error)
  • Drawing construction arcs too lightly
  • Not verifying final measurements
  • Confusing hypotenuse with other sides

• Critical Analysis Skills:

  • Identify which construction rule applies
  • Determine if triangle is possible before constructing
  • Calculate missing sides/angles when needed
  • Verify construction using multiple methods
  • Recognize special triangles (equilateral, isosceles, right)
  • Apply Pythagoras theorem correctly
  • Use angle sum property for verification

• Measurement Precision:

  • Ruler: Measure to nearest 0.1 cm
  • Protractor: Measure to nearest 1°
  • Compass: Keep consistent arc radius
  • Allow ±0.2 cm tolerance for constructed sides
  • Allow ±2° tolerance for constructed angles

Mastering triangle construction develops spatial reasoning, precision, and logical thinking - essential skills for higher mathematics and engineering!

Remember: Practice makes perfect. Construct at least 20 different triangles to master these concepts!