Similarity and Trigonometry Basics - Coordinate Geometry Applications

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Q1. In a right triangle, if one leg is 6 and the other leg is 8, what is the length of the hypotenuse?

Solution:

Using the Pythagorean theorem: c = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10.

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Q2. If the coordinates of a triangle's vertices are (0, 0), (4, 0), and (0, 3), what is the area of the triangle?

Solution:

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

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Q3. If triangle DEF is similar to triangle XYZ and the length of DE is 6 and XY is 9, 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. (6/9)² = (2/3)² = 4/9.

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Q4. In triangle DEF, if DE = 10, DF = 6, and EF = 8, which side is the longest?

Solution:

DE = 10 is the longest side since 10 > 8 and 10 > 6.

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Q5. If the radius of a circle is 5, what is the area of the circle?

Solution:

The area of a circle is given by A = πr^2. Therefore, A = π(5^2) = 25π.

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Q6. In a right triangle, if one angle is 45 degrees, what are the measures of the other two angles?

Solution:

In a right triangle, the sum of the angles is 180 degrees. Therefore, the other angle must also be 45 degrees.

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Q7. What is the perimeter of a rectangle with length 10 and width 4?

Solution:

Perimeter P = 2(length + width) = 2(10 + 4) = 2(14) = 28.

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Q8. If triangle DEF is similar to triangle XYZ and the ratio of their 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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Q9. In triangle JKL, if JK = 8, KL = 6, and JL = 10, is triangle JKL a right triangle?

Solution:

Using the Pythagorean theorem: 8² + 6² = 64 + 36 = 100 = 10², so triangle JKL is a right triangle.

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Q10. If two triangles are similar, what is the ratio of their corresponding sides if the ratio of their areas is 9:16?

Solution:

The ratio of the sides of similar triangles is the square root of the ratio of their areas. sqrt(9:16) = 3:4.