Geometry and Algebra Problems Quiz

A quadrilateral with two parallel sides and two equal non-parallel sides is called an isosceles trapezium. The opposite angles of an isosceles trapezium are supplementary. This means that if two angles are adjacent to each other, their sum will be 180 degrees. We can use this property to solve for the unknown angle.

In a rectangle, the diagonals are equal in length and bisect each other. The intersection point of the diagonals is the center of the circumcircle, which is the circle that passes through all four vertices of the rectangle.

The average of a set of numbers is the sum of the numbers divided by the total number of numbers. In this case, we have 9 numbers with an average of 22. This means the sum of these 9 numbers is 9 * 22 = 198. Similarly, the sum of the first 4 numbers is 4 * 25 = 100, and the sum of the last 4 numbers is 4 * 21 = 84. To find the 4th number, we can subtract the sum of the first 3 numbers and the last 3 numbers from the total sum, which gives us 198 - 100 - 84 = 14. Therefore, the 4th number is 14.

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. They have non-repeating, non-terminating decimal representations. The square root of a prime number is always irrational, which means √6, √5, and √7 are all irrational numbers.

The volume of a sphere is (4/3)πr³, where r is the radius of the sphere. The volume of a cube is s³, where s is the side length of the cube. To find the volume of the sphere, we need to find the radius of the sphere first. The volume of the cube is 72 cm³, which means s³ = 72 cm³. This gives us s = ∛72 cm = 4 cm. Since the sphere fits inside the cube, the diameter of the sphere is equal to the side length of the cube, which is 4 cm. Therefore, the radius of the sphere is 2 cm. Now, we can calculate the volume of the sphere: (4/3)πr³ = (4/3)π(2 cm)³ = 33.51 cm³. The volume of the enclosed sphere is approximately 33.51 cm³.