Class 7 Maths Chapter 7 Short and Long Question Answer - A Tale of Three Intersecting Lines

Q1. Can a triangle exist with side lengths 5 cm, 7 cm, and 13 cm? Verify using the triangle inequality.

Answer: To determine if a triangle exists with side lengths 5 cm, 7 cm, and 13 cm, we apply the triangle inequality: the sum of any two sides must be greater than the third side.

- Check 1: 5 + 7 = 12 < 13 (not satisfied).

- Check 2: 5 + 13 = 18 > 7 (satisfied).

- Check 3: 7 + 13 = 20 > 5 (satisfied).

- Since one condition (5 + 7 > 13) is not satisfied, a triangle cannot exist with these side lengths.

Q2. Check if a triangle exists with side lengths 6 m, 8 m, and 12 m.

Answer: Apply the triangle inequality:

Check 1: 6 + 8 = 14 > 12 (satisfied).

- Check 2: 6 + 12 = 18 > 8 (satisfied).

- Check 3: 8 + 12 = 20 > 6 (satisfied).

- All conditions are satisfied, so a triangle exists with these side lengths.

Q3: For a triangle with side lengths 3 cm, 4 cm, and 6 cm, confirm if it satisfies the triangle inequality.

Answer: The triangle inequality states that each side must be less than the sum of the other two:

- Check 1: 3 + 4 = 7 > 6 (satisfied).

- Check 2: 3 + 6 = 9 > 4 (satisfied).

- Check 3: 4 + 6 = 10 > 3 (satisfied).

Since all conditions are met, the side lengths 3 cm, 4 cm, and 6 cm satisfy the triangle inequality, confirming a triangle can exist.

Q4: Does a triangle with side lengths 2 mm, 9 mm, and 15 mm satisfy the triangle inequality?

Answer: Apply the triangle inequality:

- Check 1: 2 + 9 = 11 < 15 (not satisfied).

- Check 2: 2 + 15 = 17 > 9 (satisfied).

- Check 3: 9 + 15 = 24 > 2 (satisfied).

Since one condition fails (2 + 9 > 15), the side lengths do not satisfy the triangle inequality, and no triangle exists.

Q5: In triangle PQR, if angle P = 40 degrees and angle Q = 80 degrees, find the measure of angle R.

Answer: The angle sum property states that the sum of angles in a triangle is 180 degrees.

• Angle P + Angle Q + Angle R = 180 degrees.
• 40 + 80 + Angle R = 180.
• 120 + Angle R = 180.
• Angle R = 180 - 120 = 60 degrees.
• Thus, Angle R = 60 degrees.

Q6: In triangle XYZ, Angle X = 55 degrees and Angle Y = 65 degrees. Find the exterior angle at vertex Z.

Answer: First, find Angle Z using the angle sum property:

• Angle X + Angle Y + Angle Z = 180 degrees.

• 55 + 65 + Angle Z = 180.

• 120 + Angle Z = 180.

• Angle Z = 60 degrees.

• The exterior angle at Z is supplementary to Angle Z (they form a straight angle):

• Exterior angle = 180 - Angle Z = 180 - 60 = 120 degrees.

• Thus, the exterior angle at Z is 120 degrees.

Q7: A spider is at one corner of a rectangular box and needs to reach the opposite corner by walking on the surface. If the box dimensions are 10 cm, 12 cm, and 15 cm, calculate the length of the shortest path.

Ans: The shortest path is found by unfolding the box. Possible nets include:

• (width + height, length): (12 + 15, 10) = (27, 10).
• (length + height, width): (10 + 15, 12) = (25, 12).
• (length + width, height): (10 + 12, 15) = (22, 15).

Calculate diagonals:

• √(27² + 10²) = √(729 + 100) = √829 ≈ 28.792.
• √(25² + 12²) = √(625 + 144) = √769 ≈ 27.730.
• √(22² + 15²) = √(484 + 225) = √709 ≈ 26.627.

⇒ Shortest path is √709 ≈ 26.627 cm

Q8: In triangle LMN, if angle L = 50 degrees and angle M = 70 degrees, find the exterior angle at vertex N and determine its relationship with angles L and M. 

Ans: First, find angle N using the angle sum property: Angle L + Angle M + Angle N = 180 degrees. 
Substituting, 50 + 70 + Angle N = 180. 
This simplifies to 120 + Angle N = 180, 
so Angle N = 60 degrees. 
The exterior angle at N is supplementary to Angle N: Exterior angle = 180 - 60 = 120 degrees. 
To find the relationship, note that the exterior angle at N equals the sum of the opposite interior angles (L and M): 
50 + 70 = 120 degrees. 
Thus, the exterior angle at N is 120 degrees, and it equals the sum of angles L and M.

Q9: In triangle XYZ, the exterior angle at vertex X is 130 degrees, and angle Y = 55 degrees. Find angle Z and analyze whether the triangle is acute-angled, right-angled, or obtuse-angled based on its angles.

Ans: To determine angle Z in triangle XYZ and classify the triangle, we proceed as follows:
Calculate Angle X

• The exterior angle at X is supplementary to angle X: Angle X = 180 - Exterior angle at X.

• Substitute: Angle X = 180 - 130 = 50 degrees.

Calculate Angle Z

• Use the angle sum property: Angle X + Angle Y + Angle Z = 180 degrees.

• Substitute: 50 + 55 + Angle Z = 180.

• Simplify: 105 + Angle Z = 180.

• Solve: Angle Z = 180 - 105 = 75 degrees.

Classify the Triangle

• List angles: Angle X = 50 degrees, Angle Y = 55 degrees, Angle Z = 75 degrees.

• Definitions:
  Acute-angled: all angles < 90 degrees.
  Right-angled: one angle = 90 degrees.
  Obtuse-angled: one angle > 90 degrees.

• Check: 50, 55, 75 are all less than 90 degrees.

• Conclusion: The triangle is acute-angled.

Therefore, angle Z is 75 degrees, and the triangle is acute-angled.

Q10: In triangle UVW, if angle U = 45 degrees and angle V = 60 degrees, find the third angle W and determine the ratio of angles U, V, and W.

Ans: To find angle W in triangle UVW and the ratio of angles U, V, and W, we proceed as follows:
Calculate Angle W

• Use the angle sum property: Angle U + Angle V + Angle W = 180 degrees.

• Substitute: 45 + 60 + Angle W = 180.

• Simplify: 105 + Angle W = 180.

• Solve: Angle W = 180 - 105 = 75 degrees.

Determine Ratio of Angles

• List angles: Angle U = 45 degrees, Angle V = 60 degrees, Angle W = 75 degrees.

• Express ratio: 45 : 60 : 75.

• Find the greatest common divisor: 45, 60, 75 are divisible by 15.

• Simplify:

  • 45 / 15 = 3.

  • 60 / 15 = 4.

  • 75 / 15 = 5.

• Ratio: 3 : 4 : 5.