Basic Proportionality Theorem

🔄 Quick Recap

In the previous section, we learned about similar figures - shapes that have the same form but possibly different sizes. Now, we'll learn a very important theorem related to triangles that will help us understand when triangles are similar.

📚 The Basic Proportionality Theorem (Thales' Theorem)

The Basic Proportionality Theorem, also known as Thales' Theorem, is named after the ancient Greek mathematician Thales of Miletus who lived around 600 BCE. This theorem forms the foundation for understanding similar triangles.

Here's what the theorem states:

If a line is drawn parallel to one side of a triangle and it intersects the other two sides, then it divides those sides in the same ratio.

Let's understand this with a simple explanation:

In the triangle ABC shown above:

• DE is parallel to BC (one side of the triangle)
• DE intersects the other two sides of the triangle at points D and E
• The theorem tells us that: AD/DB = AE/EC

This means that if the line DE divides AB in the ratio 2:1, then it will also divide AC in the ratio 2:1.

🧮 Mathematical Proof

Let's see why this theorem is true. We'll use the concept of areas of triangles.

In triangle ABC with DE parallel to BC:

1. Draw triangles ADE and BDE (as shown in the diagram)
2. Triangles BDE and DEC are on the same base DE and between the same parallels DE and BC
3. Therefore, area of triangle BDE = area of triangle DEC

Now let's find the ratio of areas:

Area of triangle ADE / Area of triangle BDE = (1/2 × AD × height) / (1/2 × DB × height)
                                           = AD / DB

Similarly:

Area of triangle ADE / Area of triangle DEC = (1/2 × AE × height) / (1/2 × EC × height)
                                           = AE / EC

Since triangle BDE = triangle DEC in area, we can conclude:

AD / DB = AE / EC

And that's our Basic Proportionality Theorem proven!

🌍 The Converse of the Theorem

There's also a converse of this theorem, which states:

If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

This means if AD/DB = AE/EC, then DE is parallel to BC.

✅ Solved Examples

Example 1

In triangle ABC, DE is parallel to BC. If AD = 3 cm, DB = 2 cm, and AE = 4.5 cm, find EC.

Solution: Since DE is parallel to BC, by the Basic Proportionality Theorem:

AD/DB = AE/EC
3/2 = 4.5/EC

Cross multiply:

3 × EC = 2 × 4.5
3 × EC = 9
EC = 3 cm

Example 2

In triangle PQR, if a line divides PQ and PR in the ratio 2:3 and 4:5 respectively, is this line parallel to QR?

Solution: For the line to be parallel to QR, it should divide both sides in the same ratio.

Let's check:

PQ is divided in the ratio 2:3, which means the division ratio is 2/3
PR is divided in the ratio 4:5, which means the division ratio is 4/5

Since 2/3 ≠ 4/5, the ratios are not equal. Therefore, the line is not parallel to QR.

🧪 Activity Time!

Let's explore the Basic Proportionality Theorem with a hands-on activity:

1. Draw a large triangle on a piece of paper
2. Mark points on two sides of the triangle that divide those sides in the same ratio (for example, mark points that divide both sides in the ratio 1:2)
3. Connect those points with a straight line
4. Use a ruler to measure and verify that this line is parallel to the third side of the triangle

This activity demonstrates the converse of the Basic Proportionality Theorem!

⚠️ Common Misconceptions

• Misconception: The Basic Proportionality Theorem works for any line drawn inside a triangle. Truth: It only works for lines that are parallel to one of the sides of the triangle.

• Misconception: If a line divides one side of a triangle in a certain ratio, it will divide all sides in the same ratio. Truth: The theorem only guarantees that it divides two sides in the same ratio when the line is parallel to the third side.

💡 Key Points to Remember

• The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• The converse is also true: if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
• This theorem was discovered by Thales of Miletus, an ancient Greek mathematician.
• The theorem forms the foundation for understanding similar triangles.

🤔 Think About It!

1. What happens if we draw multiple lines parallel to one side of a triangle? How will they divide the other two sides?
2. Can you think of any real-life applications where this theorem might be useful?
3. If a line divides two sides of a triangle in the ratio 1:1 (i.e., at the midpoints), what special property does this line have?

In the next section, we'll learn about the criteria for similarity of triangles!