Name the type of the triangle in two different ways. (3)
 (i) ∆XYZ with ∠Z = 90°
 (ii) ∆ PQR such that PQ = QR = PR = 5 cm ∠L = 30°,
 (iii) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.

Let's analyze each triangle and determine its type in two different ways:

1. **(i) ∆XYZ with ∠Z = 90°**
   - **Type 1**: Right-angled triangle (since it has a 90° angle).
   - **Type 2**: Scalene triangle (assuming the sides are all different).

2. **(ii) ∆PQR such that PQ = QR = PR = 5 cm**
   - **Type 1**: Equilateral triangle (all sides are equal).
   - **Type 2**: Acute-angled triangle (all angles are less than 90°, and in an equilateral triangle, each angle is 60°).

3. **(iii) Triangle with lengths of sides 7 cm, 8 cm, and 9 cm**
   - **Type 1**: Scalene triangle (all sides are different).
   - **Type 2**: Acute-angled triangle (since the square of the longest side is less than the sum of the squares of the other two sides, meaning all angles are less than 90°).

Would you like more details or have any questions?

Here are five related questions for further practice:
1. What is the definition of an obtuse-angled triangle?
2. How can you classify triangles based on their angles?
3. Can a scalene triangle ever be equilateral?
4. What is the Pythagorean theorem, and how does it apply to right-angled triangles?
5. How do you determine if a triangle is acute, obtuse, or right-angled using side lengths?

**Tip**: For any triangle, you can classify it based on its angles (acute, right, obtuse) and its sides (scalene, isosceles, equilateral).

Name the type of the triangle in two different ways. (i) ∆XYZ with ∠Z = 90°, (ii) ∆PQR such that PQ = QR = PR = 5 cm, (iii) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.

Learn how to classify triangles by their angles and side lengths through examples: ∆XYZ with a 90° angle, equilateral ∆PQR, and a scalene triangle with sides 7 cm, 8 cm, and 9 cm. This problem involves concepts like right-angled triangles, equilateral triangles, and the Pythagorean theorem, making it ideal for Grades 6-8 students.

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