Similarity and Trigonometry Basics - Coordinate Geometry Applications

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Q1. 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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Q2. 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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Q3. If the coordinates of the vertices of a triangle are A(1, 1), B(4, 1), and C(1, 5), what is the perimeter of the triangle?

Solution:

Length AB = 3, Length AC = 4, Length BC = √((4-1)² + (1-5)²) = √(9 + 16) = 5. Perimeter = 3 + 4 + 5 = 12.

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Q4. 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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Q5. 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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Q6. If the coordinates of the vertices of a triangle are A(1, 2), B(4, 6), and C(1, 6), what is the base of the triangle AB?

Solution:

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

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Q7. 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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Q8. In triangle GHI, if GH = 8, HI = 6, and GI = 10, is triangle GHI a right triangle?

Solution:

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

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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 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.