##### Solutions from the Previous Issue:

1.        Find the perimeter of a right triangle with shorter sides measuring 5 and 12 units.

Solution:

Since the sides of a right triangle satisfy the Pythagorean Theorem, a2 + b2 = c2, we have 52 + 122 = 25 + 144 = 169 = 132. So, the three sides are 5, 12 and 13 and the perimeter is 5 + 12 + 13 = 30.

2.        Why could a right triangle with integers for lengths of sides not have shorter sides measuring 5 and 10 units?

Solution:

By the Pythagorean Theorem, 52 + 102 = 25 + 100 = 125 which is not a perfect square [125 ~ (11.1803...)2]. Thus, the third side cannot have an integer length.

3.

Observe the two squares in the figure above with side lengths of 4 and 10 units respectively. What is the area of the shaded region?

Solution:

Let H – the height of the red triangle.

;       H =2.85

4.

The four circles in the figure above each have a radius of 3 units. The centers of each circle form a square. Find the area of the green shaded region.

Solution:

Areagreen=Area square-Area circle

Areagreen= (3 + 3)2 - π∙32 = 36 – 28.26 = 7.74 units2

5.

In the above figure, the area of triangle CDE is 3 square units. Find the area of triangle ABC.

Solution:

From Pythagoras AB = = 6 units

AreaABC= square units

##### Proposed Problems

1.        The line segments joining the vertices of triangle ABC to the midpoints of the opposite edges are called medians and are concurrent at P.

Prove that P splits each median in the ratio 2:1.

2.        A second order recurrence relation is defined by un+1 = (un+un−1)/2; that is, each new term is the mean of the previous two terms.

For example, when u1=2 and u2=5, we generate the following sequence:

2, 5, 3.5, 4.25, 3.875, ... .

Find the limit of the sequence for different starting values.

3.        Four digits are selected from the set {1,2,3,4,5} to form a 4-digit number. Find the sum of all possible permutations.

4.        An arrow is formed in a 2 × 2 square by joining the bottom corners to the midpoint of the top edge and the centre of the square.

Find the area of the arrow.