Similarity and Trigonometry Basics

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Q1. If two triangles are similar with a ratio of their sides as 2:3, what is the ratio of their areas?

Solution:

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2/3)^2 = 4/9.

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Q2. If two angles of a triangle are 50 degrees and 70 degrees, what is the third angle?

Solution:

The third angle = 180 - (50 + 70) = 60 degrees.

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Q3. In triangle DEF, if DE = 5, EF = 12, and DF = 13, what type of triangle is it?

Solution:

Since 5^2 + 12^2 = 13^2 (25 + 144 = 169), triangle DEF is a right triangle.

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Q4. What is the sine of a 30-degree angle?

Solution:

The sine of 30 degrees is 0.5.

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Q5. In a circle, if the radius is 5 cm, what is the circumference?

Solution:

Circumference = 2πr = 2π(5) = 10π cm.

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Q6. In similar triangles, if the ratio of the lengths of two corresponding sides is 2:3, what is the ratio of their areas?

Solution:

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2/3)^2 = 4/9.

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Q7. In triangle JKL, if JK = 5 cm, KL = 12 cm, and JL = 13 cm, is triangle JKL a right triangle?

Solution:

By the Pythagorean theorem, 5^2 + 12^2 = 25 + 144 = 169 = 13^2, so triangle JKL is a right triangle.

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Q8. What is the cosine of a 60-degree angle?

Solution:

The cosine of 60 degrees is 0.5.

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Q9. If two triangles have sides in the ratio 3:4:5, what type of triangle are they?

Solution:

A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.

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Q10. In triangle DEF, if DE = 6 cm, DF = 8 cm, and angle E = 90 degrees, what is the length of EF?

Solution:

Using the Pythagorean theorem, EF = √(DE^2 + DF^2) = √(6^2 + 8^2) = √(36 + 64) = √100 = 10 cm.