Heron’s Formula: Geometry Quiz! Test

1. The area of a triangular signboard of sides 5 cm, 12 cm, and 13 cm is:

60 cm2

30 cm2

12 cm2

The area of a triangle can be calculated using the formula A = (base * height) / 2. In this case, the base and height of the triangle can be determined by the sides of the triangle. The side lengths given are 5 cm, 12 cm, and 13 cm. By using the Pythagorean theorem, we can determine that the triangle is a right-angled triangle. The base and height can be determined by taking the sides adjacent to the right angle. In this case, the base is 5 cm and the height is 12 cm. Plugging these values into the formula, we get A = (5 * 12) / 2 = 30 cm^2.

Explanation

The area of a triangle can be calculated using the formula A = (base * height) / 2. In this case, the base and height of the triangle can be determined by the sides of the triangle. The side lengths given are 5 cm, 12 cm, and 13 cm. By using the Pythagorean theorem, we can determine that the triangle is a right-angled triangle. The base and height can be determined by taking the sides adjacent to the right angle. In this case, the base is 5 cm and the height is 12 cm. Plugging these values into the formula, we get A = (5 * 12) / 2 = 30 cm^2.

2. The sides of a triangle are 3 cm, 4 cm, and 5 cm. Its area is:

15 cm2

12 cm2

9 cm2

6 cm2

The given triangle is a right-angled triangle with sides of lengths 3 cm, 4 cm, and 5 cm. The area of a right-angled triangle can be calculated using the formula: area = (1/2) * base * height. In this case, the base and height of the triangle are 3 cm and 4 cm respectively. Therefore, the area of the triangle is (1/2) * 3 cm * 4 cm = 6 cm2.

Explanation

The given triangle is a right-angled triangle with sides of lengths 3 cm, 4 cm, and 5 cm. The area of a right-angled triangle can be calculated using the formula: area = (1/2) * base * height. In this case, the base and height of the triangle are 3 cm and 4 cm respectively. Therefore, the area of the triangle is (1/2) * 3 cm * 4 cm = 6 cm2.

3. The sides of a triangle are in a ratio of 25:14:12 and its perimeter is 510 m. The greatest side of the triangle is:

270 m

250 m

170 m

120 m

The sides of a triangle are in a ratio of 25:14:12, which means that the lengths of the sides can be represented as 25x, 14x, and 12x, where x is a constant. The perimeter of the triangle is given as 510 m, so we can write the equation 25x + 14x + 12x = 510. Simplifying this equation gives us 51x = 510, which means x = 10. Therefore, the lengths of the sides are 250 m, 140 m, and 120 m. The greatest side of the triangle is 250 m.

Explanation

The sides of a triangle are in a ratio of 25:14:12, which means that the lengths of the sides can be represented as 25x, 14x, and 12x, where x is a constant. The perimeter of the triangle is given as 510 m, so we can write the equation 25x + 14x + 12x = 510. Simplifying this equation gives us 51x = 510, which means x = 10. Therefore, the lengths of the sides are 250 m, 140 m, and 120 m. The greatest side of the triangle is 250 m.

4. The area of ΔABC is:

20 cm2

10 cm2

Explanation

5. The perimeter of a right triangle is 60 cm and its hypotenuse is 26 cm. The other two sides of the triangle are:

26 cm, 8 cm

25 cm, 9 cm

24 cm, 10 cm

20 cm, 14 cm

In a right triangle, the sum of the lengths of the two shorter sides (legs) is equal to the length of the hypotenuse. Therefore, if the hypotenuse is 26 cm, the sum of the lengths of the other two sides must be 26 cm. Among the given options, only 24 cm and 10 cm add up to 26 cm. Hence, the correct answer is 24 cm and 10 cm.

Explanation

In a right triangle, the sum of the lengths of the two shorter sides (legs) is equal to the length of the hypotenuse. Therefore, if the hypotenuse is 26 cm, the sum of the lengths of the other two sides must be 26 cm. Among the given options, only 24 cm and 10 cm add up to 26 cm. Hence, the correct answer is 24 cm and 10 cm.

6. When the sum of squares of two sides of a triangle is equal to the square of the length of the third side, then it is called a:

Scalene triangle

Right triangle

Isosceles triangle

Equilateral triangle

A right triangle is a triangle in which one of the angles is a right angle (90 degrees). In this case, the sum of the squares of the two shorter sides of the triangle is equal to the square of the longest side, which is the definition of a right triangle according to the Pythagorean theorem. Therefore, the given statement describes a right triangle.

Explanation

A right triangle is a triangle in which one of the angles is a right angle (90 degrees). In this case, the sum of the squares of the two shorter sides of the triangle is equal to the square of the longest side, which is the definition of a right triangle according to the Pythagorean theorem. Therefore, the given statement describes a right triangle.

7. The area of the trapezium in the adjoining figure is:

286 m2

296 m2

306 m2

316 m2

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Explanation

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8. The perimeter of an equilateral triangle is 60 m. The area is:

Perimeter of an equilateral triangle = 3 × side 60 m = 3 × side Side = 60 m / 3 = 20 m Area of an equilateral triangle = (√3 / 4) × side^2 Area = (√3 / 4) × 20^2 Area = (√3 / 4) × 400 Area = 100√3 sq m

Explanation

Perimeter of an equilateral triangle = 3 × side 60 m = 3 × side Side = 60 m / 3 = 20 m Area of an equilateral triangle = (√3 / 4) × side^2 Area = (√3 / 4) × 20^2 Area = (√3 / 4) × 400 Area = 100√3 sq m

9.

4 cm

6 cm

8 cm

36 cm

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Explanation

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10. The area of the quadrilateral ABCD in the adjoining figure is:

57 cm2

95 cm2

102 cm2

114 cm2

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Explanation

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11. The area of an isosceles triangle having a base 2 cm and the length of one of the equal sides 4 cm, is

The area of an isosceles triangle can be found using the formula A = (1/2) * b * h, where b is the base and h is the height. In this case, the base is given as 2 cm and the length of one of the equal sides is 4 cm. Since the triangle is isosceles, we can divide the base into two equal parts, each measuring 1 cm. Now, we can use the Pythagorean theorem to find the height of the triangle. The height is the perpendicular distance from the midpoint of the base to the top vertex of the triangle. Using the Pythagorean theorem, we have h^2 = 4^2 - 1^2 = 15. Therefore, the height is sqrt(15) cm. Plugging these values into the formula, we get A = (1/2) * 2 * sqrt(15) = sqrt(15) cm^2.

Explanation

The area of an isosceles triangle can be found using the formula A = (1/2) * b * h, where b is the base and h is the height. In this case, the base is given as 2 cm and the length of one of the equal sides is 4 cm. Since the triangle is isosceles, we can divide the base into two equal parts, each measuring 1 cm. Now, we can use the Pythagorean theorem to find the height of the triangle. The height is the perpendicular distance from the midpoint of the base to the top vertex of the triangle. Using the Pythagorean theorem, we have h^2 = 4^2 - 1^2 = 15. Therefore, the height is sqrt(15) cm. Plugging these values into the formula, we get A = (1/2) * 2 * sqrt(15) = sqrt(15) cm^2.

12. The edges of a triangular board are 6 cm, 8 cm and 10 cm. The cost of painting it at the rate of 9 paise per cm² is:

Rs. 2.00

Rs. 2.16

Rs. 2.48

Rs. 3.00

To find the cost of painting the triangular board, we need to calculate its area first. We can use Heron's formula to find the area of a triangle when the lengths of its sides are known. The semi-perimeter of the triangle is (6+8+10)/2 = 12 cm. Using this, we can calculate the area as √(12(12-6)(12-8)(12-10)) = √(12*6*4*2) = √(576) = 24 cm^2. Multiplying this by the painting rate of 9 paise per cm^2 gives us a cost of 24 * 9 = 216 paise. Converting this to rupees gives us Rs. 2.16.

Explanation

To find the cost of painting the triangular board, we need to calculate its area first. We can use Heron's formula to find the area of a triangle when the lengths of its sides are known. The semi-perimeter of the triangle is (6+8+10)/2 = 12 cm. Using this, we can calculate the area as √(12(12-6)(12-8)(12-10)) = √(12*6*4*2) = √(576) = 24 cm^2. Multiplying this by the painting rate of 9 paise per cm^2 gives us a cost of 24 * 9 = 216 paise. Converting this to rupees gives us Rs. 2.16.

13. The area of the quadrilateral ABCD in the adjoining figure is:

14.8 cm2

15 cm2

15.2 cm2

16.4 cm2

The area of a quadrilateral can be calculated by dividing it into triangles and finding the sum of their areas. In the given figure, we can divide the quadrilateral ABCD into two triangles, triangle ABC and triangle ACD. The area of triangle ABC can be calculated using the formula for the area of a triangle, which is 1/2 * base * height. The base of triangle ABC is 4 cm and the height is 3.8 cm, so the area of triangle ABC is 7.6 cm^2. Similarly, the area of triangle ACD can be calculated as 1/2 * base * height, which is 4 cm * 3.4 cm = 13.6 cm^2. Adding the areas of the two triangles gives us a total area of 7.6 cm^2 + 13.6 cm^2 = 21.2 cm^2. However, we need to subtract the area of triangle BCD, which is 1/2 * base * height = 4 cm * 1.2 cm = 4.8 cm^2. Therefore, the area of quadrilateral ABCD is 21.2 cm^2 - 4.8 cm^2 = 16.4 cm^2.

Explanation

The area of a quadrilateral can be calculated by dividing it into triangles and finding the sum of their areas. In the given figure, we can divide the quadrilateral ABCD into two triangles, triangle ABC and triangle ACD. The area of triangle ABC can be calculated using the formula for the area of a triangle, which is 1/2 * base * height. The base of triangle ABC is 4 cm and the height is 3.8 cm, so the area of triangle ABC is 7.6 cm^2. Similarly, the area of triangle ACD can be calculated as 1/2 * base * height, which is 4 cm * 3.4 cm = 13.6 cm^2. Adding the areas of the two triangles gives us a total area of 7.6 cm^2 + 13.6 cm^2 = 21.2 cm^2. However, we need to subtract the area of triangle BCD, which is 1/2 * base * height = 4 cm * 1.2 cm = 4.8 cm^2. Therefore, the area of quadrilateral ABCD is 21.2 cm^2 - 4.8 cm^2 = 16.4 cm^2.

14.

5.196 cm2

3.496 cm2

1.732 cm2

0.866 cm2

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Explanation

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15. The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. The length of its longest altitude is:

To find the length of the longest altitude of a triangle, we can use the formula for the area of a triangle. The formula is given as: Area = (1/2) * base * height. In this case, we know the lengths of the sides of the triangle, but we need to find the height of the triangle. We can use Heron's formula to find the area of the triangle, and then divide it by the corresponding base to find the height. Using Heron's formula, we find that the area of the triangle is approximately 924. Using the formula for the area of a triangle, we can find that the length of the longest altitude is approximately 30 cm.

Explanation

To find the length of the longest altitude of a triangle, we can use the formula for the area of a triangle. The formula is given as: Area = (1/2) * base * height. In this case, we know the lengths of the sides of the triangle, but we need to find the height of the triangle. We can use Heron's formula to find the area of the triangle, and then divide it by the corresponding base to find the height. Using Heron's formula, we find that the area of the triangle is approximately 924. Using the formula for the area of a triangle, we can find that the length of the longest altitude is approximately 30 cm.