Triangles Class 10: A Complete NCERT Guide

Understanding Similarity of Triangles

What exactly does it mean for two triangles to be 'similar'? Unlike congruence, which demands equal sides and equal angles, similarity is a bit more flexible. Two triangles are said to be similar if they satisfy two key conditions:

1. Their corresponding angles are equal.
2. Their corresponding sides are in the same ratio (or proportion).

Let's consider two triangles, ΔABC and ΔPQR. If we say that ΔABC is similar to ΔPQR (written as ΔABC ~ ΔPQR), it implies:

• Angles: ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.
• Sides: AB/PQ = BC/QR = AC/PR.

The order of the vertices in the similarity notation is extremely important! It tells you exactly which angles are equal and which sides are in proportion. For example, ΔABC ~ ΔQRP would mean ∠A = ∠Q, ∠B = ∠R, and so on. Mastering this correspondence is half the battle won in solving problems related to similar triangles.

Fundamental Theorems of Triangles

Basic Proportionality Theorem (BPT) or Thales' Theorem

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Converse of BPT

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

AAA Similarity Criterion

If in two triangles, the corresponding angles are equal, then their corresponding sides are proportional, and hence the triangles are similar.

SSS Similarity Criterion

If the corresponding sides of two triangles are proportional, then their corresponding angles are equal, and hence the triangles are similar.

SAS Similarity Criterion

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the two triangles are similar.

Area of Similar Triangles Theorem

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Pythagoras Theorem

In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Worked Example: Applying the Basic Proportionality Theorem

1. Problem Statement — In a triangle ΔABC, a line segment DE is drawn parallel to the side BC, such that D is on AB and E is on AC. If AD = 2.4 cm, AE = 3.2 cm, and EC = 4.8 cm, find the length of AB.
2. Step 1: Identify the Given Information and Apply BPT — We are given that in ΔABC, DE || BC. This is the condition to apply the Basic Proportionality Theorem (Thales' Theorem). The theorem states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally. Therefore, we can write the ratio: AD / DB = AE / EC.
3. Step 2: Substitute the Known Values to Find DB — We have AD = 2.4 cm, AE = 3.2 cm, and EC = 4.8 cm. Let's substitute these values into our equation: 2.4 / DB = 3.2 / 4.8 To solve for DB, we can rearrange the equation: DB = 2.4 (4.8 / 3.2) DB = 2.4 1.5 DB = 3.6 cm.
4. Step 3: Calculate the Final Answer (AB) — The question asks for the length of the side AB. From the figure, we can see that AB is the sum of the segments AD and DB. AB = AD + DB AB = 2.4 cm + 3.6 cm AB = 6.0 cm. Final Answer: The length of AB is 6.0 cm.

Common Mistakes to Avoid in Your Exams

Pay close attention to these common pitfalls when solving problems from the Triangles chapter:

• Confusing Similarity with Congruence: Remember, congruent figures are always similar, but similar figures are not always congruent. Similarity means same shape, not necessarily the same size.
• Incorrectly Matching Corresponding Vertices: When you write ΔABC ~ ΔPQR, you are making a strong statement that ∠A=∠P, ∠B=∠Q, etc. A common error is to mismatch the vertices, like writing ΔABC ~ ΔQRP by mistake. Always match the equal angles to get the correct correspondence.
• Forgetting to Square the Ratio for Areas: The theorem for areas of similar triangles states that Area(ΔABC)/Area(ΔPQR) = (AB/PQ)². Students often forget the square and just use the ratio of the sides, leading to incorrect answers.
• Misapplying BPT: The Basic Proportionality Theorem can only be used when the line segment inside the triangle is parallel to the third side. Don't apply it otherwise. For the converse, you must prove the ratio is equal to show the lines are parallel.

Practice Questions with Solutions

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Frequently Asked Questions