**Math Labs with Activity – Ratio of the Areas of two Similar Triangle**

**OBJECTIVE**

To verify that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides

**Materials Required**

- A piece of cardboard
- A sheet of white paper
- A geometry box
- A tube of glue

**Theory**

If two triangles ABC and DEF are similar then area of ΔDEF

**Procedure**

**Step 1:** Paste the white sheet on the cardboard.

**Step 2:** Draw a ΔABC on the paper. Now, divide the side AB into four equal parts and label these points P_{1}, P_{2} and P_{3} as shown in Figure 8.2. Similarly, divide the side AC into four equal parts and label these points Q_{1}, Q_{2} and Q_{3} as shown in Figure 8.2. Also, divide the side BC into equal parts and label these points R_{1}, R_{2} and R_{3} as shown in Figure 8.2.

**Step 3:** Join these points to form the line segments P_{1}Q_{1}, P_{2}Q_{2} and P_{3}Q_{3} parallel to the side BC, the line segments Q_{1}R_{1}, Q_{2}R_{2} and Q_{3}R_{3} parallel to the side AB, and also the line segments P_{3}R_{1}, P_{2}R_{2} and P_{1}R_{3} parallel to the side AC. Thus, the ΔABC is divided into 16 smaller triangles (see Figure 8.2).

**Step 4:** Draw another triangle DEF having sides DE = ¾ AB, DF = ¾ AC and EF = ¾ BC. Then, clearly the ΔDEF will be similar to the ΔABC.

**Step 5:** Divide the side DE into three equal parts and label these points X_{1} and X_{2} as shown in Figure 8.3.

Divide the side DF into three equal parts and label these points Y_{1} and Y_{2} as shown in Figure 8.3.

Also, divide the side EE into three equal parts and label these points Z_{1} and Z_{2} as shown in Figure 8.3.

**Step 6:** Join these points to form the line segments X_{1}Y_{1} and X_{2}Y_{2} parallel to the side EE, the line segments Y_{1}Z_{1} and Y_{2}Z_{2} parallel to the side DE, and also the line segments X_{1}Z_{2} and X_{2}Z_{1} parallel to the side DF. Thus, the ΔDEE is divided into 9 smaller triangles (see Figure 8.3).

**Observations**

- ΔABC and ΔDEF are similar to each other.
- ΔABC is divided into 16 smaller triangles, all congruent to each other. Therefore, all these 16 triangles are equal in area.
- ΔDEF is divided into 9 smaller triangles, all congruent to each other. Therefore, all these 9 triangles are equal in area.
- Each small triangle within the ΔABC is congruent to each small triangle within the ΔDEF. Therefore, all these 25 triangles are equal in area (say, equal to M).

**Calculations**

**Result**

It is verified that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides.

**Remarks:**

The above result can also be proved for

- other pairs of corresponding sides of the two triangles.
- other triangles by dividing each side of a triangle into 3, 4,5 or 6 or even more parts and forming small triangles, and then taking a part of this triangle as a similar triangle.

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