Sum of Consecutive Numbers

Problem 1 Can you work out the answer to this number pattern and explain why you think you have the correct answer? You probably observed that the sum is the square of the middle number. Can you work out why? Why do you have a square number for the sum? Solution 1 Solution 2 Solution [...]

Original by Erlina Ronda

Geometric Interpretation of Trapezoid Area Formula

Problem 1 Show that the area of a trapezoid (trapezium) is half its height times the sum of the parallel sides. Solution Cut the trapezoid along the midpoints of the pair of non-parallel side. This halves its height. Rotate one of the parts 180 degrees about a midpoint. In the figure below, it’s about the [...]

Original by Erlina Ronda

Proving trigonometric identities #2

Problem: Prove that the equation below is an identity: 2 sin2Ө – sin4Ө = (1 – cos2Ө)(1 + cos2Ө). Solution To prove that an equation is an identity, one must show that the Left Hand Side (LHS) of the equation is identical to its Right Hand Side (RHS). That is, LHS ≡ RHS. Proof LHS: 2 sin2Ө [...]

Original by Erlina Ronda

Proof of Triangle Exterior Angle Theorem

The exterior angle of a triangle is the angle that forms a linear pair with an interior angle of the the triangle. To draw the exterior angle all you need to do is to extend the side of the triangle. A linear pair is a pair of angle that forms a straight line. In the figure [...]

Original by Erlina Ronda

Three proofs that the sum of angles of a triangle is 180

Theorem. The sum of the interior angles of any triangle is 180°. Here are three proofs for the sum of angles of triangles. Proof 1 uses the fact that the alternate interior angles formed by a transversal with two parallel lines are congruent. Proof 2 uses the exterior angle theorem. Proof 3 uses the idea [...]

Original by Erlina Ronda

What is Conditional Probability?

The conditional probability of an event B is the probability that the event will happen given the knowledge that an event A has already happened. This probability is written P(B|A), notation for the probability of B given A. In the case where events A and B are independent, the conditional probability of event B given event A is simply the probability of event B, that is P(B). If events A and B are not independent, then the probability [...]

Original by Erlina Ronda

What is cos b?

Problem: What is cos b equal to? Solution: Explanation: Bill Cosby

Original by Erlina Ronda

How to transform a pentagon into a triangle of the same area

This is the third in the series of problems involving transforming polygons into another polygon where the area is preserved. The first is about transforming a shape into another shape of the same area and the second is about transforming a quadrilateral into a triangle of the same area. The following problem builds on the [...]

Original by Erlina Ronda

How to transform quadrilaterals into triangles with the same area

Quadrilaterals have four sides. To make triangles out of them, three of the vertices should be collinear. Now, what construction should you make so that the area remain unchanged? Here’s how you can transform quadrilaterals into triangles with the same are in three easy steps. 1. Extend one of the sides of the quadrilateral. Let’s [...]

Original by Erlina Ronda

Transforming shapes without changing their areas

Under any of Euclidean transformations—reflection, rotation, and translation, the shape of a geometric object will not change; only the position and orientation of the object will change. That is, the object and its transformation are congruent. How about when the shape can change but the area remain unchanged? Try this problem.   Problem How do you [...]

Original by Erlina Ronda