1. What is the Pythagoras Theorem?

The Pythagoras Theorem states:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Mathematically: \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse and \(a, b\) are the other two sides.

2. What is the Basic Proportionality Theorem (Thales Theorem)?

The Basic Proportionality Theorem states:
If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those sides in the same ratio.

Mathematically: \(\frac{AP}{PB} = \frac{AQ}{QC}\).

3. How do you prove two triangles are similar?

Two triangles are similar if:

• AA (Angle-Angle) Similarity: Two angles of one triangle are equal to two angles of another triangle.
• SAS (Side-Angle-Side) Similarity: One angle of a triangle is equal to one angle of another triangle, and the sides including these angles are in proportion.
• SSS (Side-Side-Side) Similarity: The corresponding sides of two triangles are in the same ratio.

4. What is the formula for the area of similar triangles?

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Mathematically: \(\frac{\text{Area of } \triangle 1}{\text{Area of } \triangle 2} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2\).

5. Solve: A triangle has sides 6 cm, 8 cm, and 10 cm. Verify if it is a right-angled triangle.

Solution:

1. Check if \(c^2 = a^2 + b^2\), where \(c\) is the longest side (10 cm).
2. \(10^2 = 6^2 + 8^2\).
3. \(100 = 36 + 64\).
4. \(100 = 100\).

Yes, it is a right-angled triangle.

6. Solve: In a triangle, a line parallel to one side divides the other two sides into segments of lengths 3 cm, 5 cm, 4.5 cm, and 7.5 cm. Verify the Basic Proportionality Theorem.

Solution:

1. Lengths on one side: 3 cm and 5 cm.
2. Lengths on the other side: 4.5 cm and 7.5 cm.
3. Calculate ratios: \(\frac{3}{5} = 0.6\) and \(\frac{4.5}{7.5} = 0.6\).

Ratios are equal, so the Basic Proportionality Theorem is verified.

7. Solve: Find the ratio of areas of two similar triangles if their corresponding sides are in the ratio 2:3.

Solution:

1. Use the formula: \(\frac{\text{Area of } \triangle 1}{\text{Area of } \triangle 2} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2\).
2. \(\frac{\text{Area of } \triangle 1}{\text{Area of } \triangle 2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}\).

The ratio of the areas is 4:9.