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Triangle Proportionality Theorem

By Shyama Chandran

Learn about what the Triangle Proportionality Theorem is and how to prove it. Find the theorem, example problems, and practice problems at Thinkster Math.

Why is this concept useful?

The Triangle Proportionality Theorem is useful in calculating the length in which the sides of a triangle are divided by a line which is drawn parallel to the third side of the triangle

Where does this concept fit into the curriculum?

High School Math

What is Triangle Proportionality Theorem?

The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides the two sides proportionally.

How to Prove the Triangle Proportionality Theorem?

Let us consider a triangle △ABC

A line DE parallel to the side BC intersects the sides AB and AC.

Thus, if DE // BC, prove that AD/DB = AE/EC

For the parallel lines DE and BC, the line AB is a transversal intersecting both lines.

The ∠ADE and ∠ABC are corresponding angles and thus congruent.

Similarly, for the parallel lines DE and BC, the line AC is a transversal intersecting both lines.

The ∠AED and ∠ACB are corresponding angles and thus congruent.

∠EAD is common to both △ABC and △ADE.

Therefore, △ABC is similar to △ADE.

Which means, AD/AB = AE/AC

AD/(AD + DB) = AE/(AE + EC)

AD (AE + EC) = AE (AD + DB)

AD(AE) + AD(EC) = AE(AD) + AE(DB)

AD(AE) + AD(EC) = AE(AD) + AE(DB)

AD(EC) = AE(DB)

AD/DB = AE/EC


Sample Math Problems

Question 1: In the figure given below, calculate the value of x.

Solution:

As DE // BC, as per the Triangle Proportionality Theorem,

AD/DB = AE/EC

5/7 = x/(x+6)

5 (x + 6) = 7x

5x + 30 = 7x

7x – 5x = 30

2x = 30

x = 15cm (Ans)



Question 2: In the figure given below, calculate the length of the side AC.

Solution:

As DE // BC, as per the Triangle Proportionality Theorem,

AD/DB = AE/EC

11/17 = x(x+36)

11 (x + 36) = 17x

11x + 396 = 17x

17x - 11x = 396

6x = 396

x = 66cm

Length of AC = AE + EC = x + x + 36 = 2x + 36 = 2 X 66 + 36 = 168 cm (Ans)


Question 3: In the figure given below, calculate the length of the side AC.

Solution:

As DE // BC, as per the Triangle Proportionality Theorem,

AD/DB = AE/EC

24/8 = x/19

x/8= 24 X 19/8 = 456/8

x = 57cm

Length of AC = AE + EC = x + 19 = 57+19 = 76cm (Ans)

Question 4: In the figure given below, calculate the value of x.

Solution:

As DE // BC, as per the Triangle Proportionality Theorem,

AD/DB = AE/EC

15/21= x/(x+4)

15 (x + 4) = 21x

15x + 60 = 21x

21x – 15x = 60

6x = 60

x = 10 cm (Ans)


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Practice Math Problems

Question 1: In the figure given below, calculate the value of x.

Question 2: In the figure given below, calculate the length of AC.

Question 3: In the figure given below, calculate the value of x.


Question 4: In the figure given below, calculate the length of the side AC.

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