Math Labs with Activity - Proportionality Theorem, or Thales Theorem

Math Labs with Activity – Proportionality Theorem, or Thales Theorem

OBJECTIVE

To verify the basic proportionality theorem, or Thales theorem

Materials Required

1. A sheet of whitepaper
2. A sheet of colored paper
3. A piece of cardboard
4. A tube of glue
5. A ruler
6. A geometry box

Theory
Basic proportionality theorem (or Thales theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides then the line divides these sides in the same ratio.

Procedure
Step 1: Paste the sheet of white paper on the cardboard.
Step 2: Draw a straight line PQ = 10 cm on this paper. At P draw a line MP perpendicular to PQ and at Q draw a line NQ perpendicular to PQ. Graduate the two lines MP and NQ as shown in Figure 6.1.
Step 3: Draw a line XY parallel to PQ by taking points X and Y at an equal distance from P and Q respectively, and then joining them.
Step 4: Cut a triangle ABC from the colored paper and paste it on the white paper so that its side BC coincides with the line XY as shown in Figure 6.1.
Step 5: Draw two straight lines X₁Y₁ and X₂Y₂ both parallel to the line PQ (by the procedure discussed in Step 3), such that the line X₁Y₁ cuts the sides AB and AC at points D₁ and E₁ respectively, and the line X₂Y₂ cuts the sides AB and AC at points D₂ and E₂ respectively.
Step 6: Measure the lengths of the line segments AD₁, D₁B, AE₁, E₁C, AD₂, D₂B, AE₂ and E₂C, and record them in the observation table.
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Observations and Calculations

Result
We find that

Hence, the basic proportionality theorem, or Thales theorem, is verified.

Remarks: It is important to note that since X₁Y₁||PQ,  X₂Y₂||PQ and  XY||PQ, therefore both the lines D₁E₁ and D₂E₂ are parallel to the side BC of the triangle ABC.

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