Detailed, step-by-step solutions for DAV Class 7 Secondary Mathematics Chapter 10: Construction of Triangles (Brain Teasers, HOTS & Value Based Questions). Includes complete construction procedures, geometric justifications, angle sum calculations, Higher Order Thinking Skills (HOTS), and Value-Based Questions (VBQs) strictly matching the DAV Education curriculum.
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Q.1. (A) Tick (✓) the correct option
(i) A closed plane figure bounded by three straight line segments having three angles is known as a:
- (a) Square
- (b) Rectangle
- (c) Triangle
- (d) Pentagon
Correct Option: (c) Triangle
(ii) What are the possible dimensions for constructing a triangle of perimeter 5 metres?
- (a) 1 m, 1 m, 3 m
- (b) 1 m, 2 m, 2 m
- (c) 5 m, 1.5 m, 3 m
- (d) 1 m, 1.5 m, 2.5 m
Perimeter must equal 5 m, and the Triangle Inequality Theorem requires that the sum of any two sides must be strictly greater than the third side (a + b > c).
• In (a): 1 + 1 = 2 ≯ 3 (Cannot form a triangle).
• In (b): Perimeter = 1 + 2 + 2 = 5 m. Checks: 1 + 2 = 3 > 2 (True); 2 + 2 = 4 > 1 (True). Valid!
• In (d): 1 + 1.5 = 2.5 ≯ 2.5 (Sum equals third side, so it collapses into a line).
Correct Option: (b) 1 m, 2 m, 2 m
(iii) Which triangle criterion is used when a right angle, hypotenuse, and one side are given?
- (a) SSS
- (b) SAS
- (c) ASA
- (d) RHS
Correct Option: (d) RHS
(iv) Can a triangle be constructed with angle measures 70°, 60°, and 60°?
- (a) Yes
- (b) No
- (c) Only if equilateral
- (d) Cannot be determined
Correct Option: (b) No
Q.1. (B) Short Answer Conceptual Questions
(a) What is the minimum number of independent measurements required to construct a unique triangle?
Ans: 3 independent elements (such as 3 sides [SSS], 2 sides and included angle [SAS], 2 angles and 1 side [ASA], or RHS).
(b) Why can a triangle never have two right angles or two obtuse angles?
Ans: If a triangle had two right angles (90° + 90° = 180°), the sum of just two angles would reach 180°, leaving 0° for the third angle. If it had two obtuse angles (> 90° each), their sum would exceed 180°, violating the Angle Sum Property.
(c) Is it possible to construct a triangle with side lengths 3 cm, 4 cm, and 7 cm?
Ans: No. Sum of the two smaller sides is 3 cm + 4 cm = 7 cm, which is equal to (not strictly greater than) the third side. Hence, no triangle can be formed.
(d) In a right-angled triangle, which side is always the longest?
Ans: The Hypotenuse (the side opposite to the 90° right angle).
Q.2. Construct ΔABC in which ∠A = 65°, ∠B = 75°, and BC = 4 cm
The base given is BC = 4 cm. Therefore, we need the angles at vertices B and C to construct the triangle on base BC.
We know: ∠A + ∠B + ∠C = 180°
⇒ 65° + 75° + ∠C = 180°
⇒ 140° + ∠C = 180°
⇒ ∠C = 180° − 140° = 40°.
Steps of Construction:
1. Draw a base line segment BC = 4 cm using a ruler.
2. At point B, draw a ray BX making an angle ∠CBX = 75° using a protractor.
3. At point C, draw a ray CY making an angle ∠BCY = 40° using a protractor.
4. Let the two rays BX and CY intersect at point A.
5. ΔABC is the required triangle. (You can verify by measuring ∠A = 65°).
Construction Complete: ΔABC verified
Q.3. Construct a right-angled ΔPQR in which ∠Q = 90°, QR = 4.5 cm, and Hypotenuse PR = 6 cm (RHS Criterion)
1. Draw a line segment QR = 4.5 cm using a ruler.
2. At vertex Q, construct a right angle (∠RQX = 90°) using a compass and ruler (or protractor).
3. With R as centre and a radius of 6 cm (hypotenuse), draw an arc using a compass to cut the perpendicular ray QX at point P.
4. Join point P to point R (PR = 6 cm).
5. ΔPQR is the required right-angled triangle.
Construction Complete: Right ΔPQR verified
Q.4. Construct an equilateral triangle of perimeter 15 cm
An equilateral triangle has all 3 sides equal in length.
Side length = (Perimeter) ÷ 3 = 15 cm ÷ 3 = 5 cm.
Steps of Construction (SSS Criterion):
1. Draw a base line segment AB = 5 cm.
2. With A as centre and radius = 5 cm, draw an arc above AB.
3. With B as centre and same radius = 5 cm, draw another arc cutting the previous arc at point C.
4. Join AC and BC.
5. ΔABC is the required equilateral triangle with side 5 cm and perimeter 15 cm.
Construction Complete: Equilateral ΔABC verified
🔥 Higher Order Thinking Skills (HOTS)
HOTS Q1: The angles of a triangle are in the ratio 2 : 3 : 5. Can a triangle be constructed with these angles? If yes, classify the triangle.
Let the angles be 2x, 3x, and 5x.
By Angle Sum Property: 2x + 3x + 5x = 180°
⇒ 10x = 180° ⇒ x = 18°.
Angles are:
• First angle = 2 × 18° = 36°
• Second angle = 3 × 18° = 54°
• Third angle = 5 × 18° = 90°
Conclusion: Yes, the triangle can be uniquely constructed. Since one angle is exactly 90°, it is a Right-Angled Triangle (specifically a Scalene Right Triangle).
Ans: Yes, Right-Angled Triangle (36°, 54°, 90°)
HOTS Q2: An architect needs to construct a triangle with perimeter 18 cm whose sides are in the ratio 3 : 4 : 5. Find the side lengths and verify if it forms a right triangle.
Perimeter = 3k + 4k + 5k = 12k = 18 cm
⇒ k = 18/12 = 1.5 cm.
Sides are:
• a = 3 × 1.5 = 4.5 cm
• b = 4 × 1.5 = 6 cm
• c = 5 × 1.5 = 7.5 cm
Pythagorean Check:
a2 + b2 = (4.5)2 + (6)2 = 20.25 + 36 = 56.25.
c2 = (7.5)2 = 56.25.
Since a2 + b2 = c2, it forms a Right-Angled Triangle.
Ans: Sides: 4.5 cm, 6 cm, 7.5 cm (Right Triangle)
🌱 Value Based Questions (VBQ)
VBQ: Students of an Eco-Club plan a triangular herbal garden in their school. They mark base BC = 6 metres and construct angles of 60° and 60° at the corners B and C to enclose the maximum symmetrical area safely.
(a) What kind of triangular garden is formed and what will be the third corner angle?
Third angle = 180° − (60° + 60°) = 60°.Since all three angles are 60°, an Equilateral Triangle is formed with all sides equal to 6 metres.
(b) What environmental value is reflected by the Eco-Club members?
Core Values Depicted: Environmental stewardship, botanical conservation, teamwork, and practical application of geometry for public green spaces.Ans: (a) Equilateral Triangle (60°) | (b) Environmental Preservation & Teamwork
📚 DAV Class 7 Maths Chapter 10: Construction of Triangles – Complete Solutions Index
| Worksheet / Section | Criteria & Focus | Solution Link |
|---|---|---|
| Worksheet 1 | SSS Criterion (Three Sides Given) | View Worksheet 1 → |
| Worksheet 2 | SAS Criterion (Two Sides and Included Angle) | View Worksheet 2 → |
| Worksheet 3 | ASA / AAS Criterion (Two Angles & Side) | View Worksheet 3 → |
| Worksheet 4 | RHS Criterion (Right Angle, Hypotenuse & Side) | View Worksheet 4 → |
| Brain Teasers, HOTS & VBQs | Comprehensive Review, Angle Sum Proofs & Real Applications | Current Page |