**Math Labs with Activity – Proportionality Theorem, or Thales Theorem**

**OBJECTIVE**

To verify the basic proportionality theorem, or Thales theorem

**Materials Required**

- A sheet of whitepaper
- A sheet of colored paper
- A piece of cardboard
- A tube of glue
- A ruler
- 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 X_{1}Y_{1} and X_{2}Y_{2} both parallel to the line PQ (by the procedure discussed in Step 3), such that the line X_{1}Y_{1} cuts the sides AB and AC at points D_{1} and E_{1} respectively, and the line X_{2}Y_{2} cuts the sides AB and AC at points D_{2} and E_{2} respectively.

**Step 6:** Measure the lengths of the line segments AD_{1}, D_{1}B, AE_{1}, E_{1}C, AD_{2}, D_{2}B, AE_{2} and E_{2}C, 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 X_{1}Y_{1}||PQ, X_{2}Y_{2}||PQ and XY||PQ, therefore both the lines D_{1}E_{1} and D_{2}E_{2} are parallel to the side BC of the triangle ABC.

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