A Greedy person is celebrating his birthday with 6 of his friends. His mother baked him a birthday cake in the shape of a regular hexagon. Wanting to keep most of the cake, he makes cuts linking the midpoints of every 2 adjacent sides, and distributes these 6 slices to his friends. What proportion of the cake does he have left for himself?
A regular hexagon is made of six congruent triangles of equal sides. Let L = side of the triangle. The area can be calculated as follows: Area of initial hexagon = 6 * ( L * SQRT3 /2 L / 2 ) = 3/2 SQRT3* L^2 By linking the midpoints of two adjacent sides a smaller regular hexagon is formed of side equal to SQRT3/2 * L Area of smaller hexagon = 3/2 SQRT3 * (SQRT3/2 * L) ^2 = 9/8 SQRT3 * L^2 By dividing the two areas, we obtain: Area of smaller hexagon / area of initial hexagon = 9/8 SQRT3* L^2 / 3/2 SQRT3* L^2 = 3/4 The area of the remaining cake is 3/4 of the initial cake
A point inside a pentagon is connected to the midpoints of the sides of the pentagon and to a corner, as shown in the figure. The areas of the six regions are written in the figure.
What is the value of A – B?
A boy started from one corner of a rectangular path and walks along the adjacent sides to reach opposite corner. If he had walked through a shortest distance diagonally, he could have saved the distance equal to five-sixth of the shortest side a park, then what is the ratio of a longest side to the shortest side?
If the length of two sides of a right angled triangle measured in inches are prime numbers, and it has two adjacent sides of 12 and 13 inches long, how long must the third side be?
The lengths of two adjacent sides of a parallelogram are 8 and 15, respectively. Find the sum of squares of the lengths of the two diagonals.
