Similarity and Trigonometry Basics - Proof-based Questions

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Q1. In coordinate geometry, what is the distance between the points (1, 2) and (4, 6)?

Solution:

Distance = √((4-1)² + (6-2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

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Q2. In triangle PQR, if PQ = 12 cm, PR = 16 cm, and QR = 20 cm, is triangle PQR similar to triangle ABC with sides 3 cm, 4 cm, and 5 cm?

Solution:

The sides of triangle PQR are in the ratio 12:16:20 = 3:4:5, which matches triangle ABC, confirming similarity.

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Q3. If triangle GHI is similar to triangle JKL and the sides of triangle GHI are 3 cm, 4 cm, and 5 cm, what is the ratio of the sides of triangle JKL if the longest side is 10 cm?

Solution:

The ratio of the longest sides is 10 cm to 5 cm, which simplifies to 2:1. Therefore, the sides of triangle JKL are in the ratio 2:1.

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Q4. If the coordinates of points A and B are (2, 3) and (5, 7) respectively, what is the distance AB?

Solution:

Distance AB = √((5-2)² + (7-3)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

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Q5. In triangle MNO, if MN = 12 cm, NO = 16 cm, and MO = 20 cm, is triangle MNO a right triangle?

Solution:

Using the Pythagorean theorem, 12² + 16² = 144 + 256 = 400 = 20², so triangle MNO is a right triangle.

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