Tag: locus of points

Posted in Algebra

A challenging complex number problem with solution

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This complex number problem was selected on the basis of its uniqueness in terms of phrasing things within the Argand diagram/locus context. While my proposed solutions might be short, bear in mind this question truly demands/challenges the student to think unconventionally in order to formulate a viable solving approach.

 

Problem

A complex number z=x+iy is represented by the point P in an Argand diagram. If the complex number w where w = \frac{z-8i}{z+6}, (z\neq-6) has its real part zero, show that the locus of P in the Argand diagram is a circle and find the radius and the coordinates of the centre of this circle. If, however, w is real, find the locus of P in this case.

Solution

complex number problem

The author of this post is Mr. Frederick Koh. He is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com.

If you love this problem, I’m sure you will also enjoy the two other challenging problems shared by Mr. Koh in this site:

  1. Differentiation in parametric context
  2. Working with summation problems
Posted in Geometry

Twelve definitions of a square

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How does mathematics define a math concept?

Definitions of concepts in mathematics are different from definitions of concepts in other discipline or subject area. A definition of a concept in mathematics give a list of properties of that concept. A mathematics object will only be an example of that concept if it fits ALL those requirements, not just most of them. Further, a definition is also stated in a way that the concept being defined belongs to an already ‘well-defined’ concept. On top of this, economy of words and symbols and properties are highly observed.

Does a math concept only have one definition? Of course, not. A concept can be defined in different ways, depending on your knowledge about other math objects. In a study by Zaskin and Leikin, they suggested that the definitions students give about a concept mirrors their knowledge of mathematics. Below are examples of definitions of squares from that research. Do you think they are all legitimate definitions?

What is a square?

A square is

  1. a regular polygon with four sides
  2. a quadrilateral with all the angles and all the sides are equal
  3. a quadrilateral with all the sides equal and an angle of 90 degrees
  4. a rectangle with equal sides
  5. a rectangle with perpendicular diagonals
  6. a rhombus with equal angles
  7. a rhombus with equal diagonals
  8. a parallelogram with equal adjacent angles and equal adjacent sides
  9. a parallelogram with equal and perpendicular diagonals
  10. a quadrilateral having 4 symmetry axes
  11. a quadrilateral symmetric under rotation by 90 degrees
  12. the locus of all the points in a plane for which the sum of the distances from two given perpendicular lines is constant. Click this link to visualize #12.

 

Making (not stating) definitions is a worthwhile assessment task.

Here’s three great references for definitions of mathematical concepts. The first is from no other than Dr. Math (The Math Forum Drexel University). The middle one’s for mom and kids – G is for Google and the third’s a book of definitions for scientists and engineers.

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Posted in Geogebra, Geometry

Problem on proving perpendicular segments

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This problem is a model created to solve the problem posed in the lesson Collapsible.

In the figure CF = FB = FE. If C is moved along CB, describe the paths of F and E. Explain or prove that they are so.

This problem can be explored using GeoGebra applet.  Click this link to explore before you read on.

perpendicular segments

One way to prove that FC is a straight line and perpendicular to AC is to show that FC is a part of a right triangle. To do this to let x be the measure of FCB. Because FCB is an isosceles triangle, FBC and CFB is (180-2x).  This implies that EFB is 180-2x being supplementary to CFB thus CFB must be 2x. Triangle EFB is an isosceles triangle so FBC must be (180-2x)/2. Adding CFB and FBC we have x+ (180-2x)/2 which simplifies to 90. Thus, EB is perpendicular to CB.

The path of F of course is circular with FB as radius.