Basic Proportionality Theorem

Objectives:

1. Familiarity with idea of congruence, similarity and similar triangles
2. Visualizing BPT - If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
3. Logical proof of BPT
4. Basic understanding of Trigonometry – ratio of sides in a right triangle

Pre-requisites

1. Prior knowledge of triangle and prior objects (segment/ray, line, angle)
2. Right triangle – Pythagoras theorem

Session plan:

1. Congruence
   1. Segment, angle, triangle, quadrilateral, odd shaped figures
   2. Measures of corresponding sides and angles of congruent polygons will be equal
2. Similarity
   1. Any circle is similar to any other circle.
   2. Same holds for Square - https://geogebra.org/m/ceapgrs5
   3. 
      

Download this geogebra file from this link.

1. 1. and Equilateral Triangles and https://geogebra.org/m/kpww6afy
   2. Quadrilaterals
      1. Two quadrilaterals of the same number of sides are similar, if
         1. (i) their corresponding angles are equal and
         2. (ii) their corresponding sides are in the same ratio (or proportion)
   3. Triangle - https://geogebra.org/m/mdc43fbt
      1. if all angles of one are congruent with the corresponding angles of the second (AAA)
      2. if the ratio of three corresponding sides are equal (SSS)
2. Concept of height of a triangle. https://geogebra.org/m/k56qc3hm
   1. The height of a triangle will be inside the triangle (acute angled triangle), outside the triangle (obtuse angled triangle) and on the side of the triangle (right triangle)
   2. Selection of side as base can change, but area (half * base *height) does not change
3. BPT - If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
   1. Draw few triangles and check that this is true – visual proof https://geogebra.org/m/nctk4smk
   2. Logical Proof of BPT - https://geogebra.org/m/pjdj65cd

Process

   1. Congruent and similar triangle - 5. Exploring similarity - changing side lengths and scaling sides.ggb -  

       1. Show visual proof of congruence and CPCT.

       2. Show visual proof of similarity and implication for sides (ratio of corresponding sides = and corresponding angles =)

       3. Show for a right angle, if we take another angle, the ratio of opposite side to hypotenuse will be = always, ratio of adjacent angle to hypotenuse and opposite side to adjacent side = always.

   2. 1. Equilateral, Isosceles triangles - Side and angle measure relationships.ggb - Concept of Isosceles triangle – two sides are equal and the two angles opposite these two sides are equal

   3. Shadow cast by sun on stick – arrive at the height of the stick based on length of shadow (angle = 45)

   4. Explain concept of tan, sin and cos

   5. Explain angle of elevation and angle of depression (text book images also)

   6. 3. sin cos tan with sliders for angle and side measures.ggb

       1. Show geogebra file explaining calculation of sin, cos and tan – using slider for angle values.

       2. Show Sin, Cos, Tan values for different angles

       3. Show even if side measure changes, the angles and hence ratios do not change

   7. Show Geogebra file mnemonic file for memorizing formulae for Sin, Cos and Tan

   8. Show workbook to show the pattern to memorize sin, cos, tan of common angles

Evaluation:

Slides for rapid questions, with images