Chapter 6: Triangles - Board Exam Notes

Exam Weightage & Blueprint

Total: 7-9 Marks

This chapter is part of the Geometry Unit. Focus heavily on theorem proofs and similarity criteria.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	High	Basic Proportionality Theorem (BPT) Applications
Short Answer	2 or 3	Medium	Similarity Criteria (AA, SAS, SSS)
Long Answer	5	High	Proof of BPT or Similarity Theorems

⏰ Last 24-Hour Checklist

Congruent vs Similar: Similar = same shape, diff size.
BPT (Thales Theorem): $\frac{AD}{DB} = \frac{AE}{EC}$ if $DE || BC$.
Similarity Criteria: AAA, AA, SSS, SAS.

Area Ratio Theorem: Ratio of areas = Ratio of squares of corresp. sides (Deleted in some boards, check syllabus).
Shadow Problems: Use similarity of triangles formed by sun rays.

Theorems & Proofs ★★★★★

Basic Proportionality Theorem (BPT/Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Proof of BPT (Theorem 6.1)

Given: $\triangle ABC$, $DE || BC$.

To Prove: $\frac{AD}{DB} = \frac{AE}{EC}$.

1. Area($\triangle ADE$) = $\frac{1}{2} \times AD \times EN$

2. Area($\triangle BDE$) = $\frac{1}{2} \times DB \times EN$

3. Ratio 1: $\frac{\text{ar}(ADE)}{\text{ar}(BDE)} = \frac{AD}{DB}$

4. Similarly Ratio 2: $\frac{\text{ar}(ADE)}{\text{ar}(DEC)} = \frac{AE}{EC}$

5. Since $\triangle BDE$ and $\triangle DEC$ are on same base $DE$ and between same parallels $BC$ and $DE$, their areas are equal.

6. Therefore, $\frac{AD}{DB} = \frac{AE}{EC}$.

Similarity Criteria 🔥🔥🔥

AAA / AA

If corresponding angles are equal, triangles are similar.

SSS

If corresponding sides are proportional, triangles are similar.

SAS

One angle equal and sides including it are proportional.

Note: AA is sufficient for similarity. You don't need to prove all three angles.

Solved Examples (Board Marking Scheme)

Q1. In $\triangle ABC$, $DE || BC$. If $AD=1.5$ cm, $DB=3$ cm, $AE=1$ cm, find $EC$. (2 Marks)

Step 1: State Theorem 0.5 Mark

Since $DE || BC$, by Basic Proportionality Theorem:

$\frac{AD}{DB} = \frac{AE}{EC}$

Step 2: Substitute Values 1 Mark

$\frac{1.5}{3} = \frac{1}{EC}$

$\frac{1}{2} = \frac{1}{EC}$

Step 3: Solve 0.5 Mark

$EC = 2$ cm.

Q2. Diagonals of a trapezium ABCD with $AB || DC$ intersect at O. Show that $\frac{AO}{BO} = \frac{CO}{DO}$. (3 Marks)

Step 1: Identify Triangles 1 Mark

Consider $\triangle AOB$ and $\triangle COD$.

$\angle AOB = \angle COD$ (Vertically opposite angles)

Step 2: Alternate Angles 1 Mark

Since $AB || DC$, alternate interior angles are equal:

$\angle OAB = \angle OCD$

Step 3: Apply Similarity 1 Mark

By AA criterion, $\triangle AOB \sim \triangle COD$.

Therefore, corresponding sides are proportional:

$\frac{AO}{CO} = \frac{BO}{DO} \Rightarrow \frac{AO}{BO} = \frac{CO}{DO}$ (Rearranging terms)

Previous Year Questions (PYQs)

2023 (1 Mark): If $\triangle ABC \sim \triangle DEF$ and $AB=3, DE=4.5$, find ratio of areas.
Ans: Ratio of areas = Square of ratio of sides = $(3/4.5)^2 = (2/3)^2 = 4:9$.

2020 (3 Marks): A vertical pole of length 6m casts a shadow 4m long. At the same time, a tower casts a shadow 28m long. Find height of tower.
Ans: $\triangle ABC \sim \triangle PQR$ (Sun's elevation is same). $\frac{6}{h} = \frac{4}{28} \Rightarrow h = 42$ m.

2019 (4 Marks): Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Ans: Standard theorem proof. Draw altitudes, use area formula and similarity.

Exam Strategy & Mistake Bank

Mistake Bank 🚨

Order of Vertices: Writing $\triangle ABC \sim \triangle DEF$ when actually $\triangle ABC \sim \triangle EDF$. Order matters!

BPT vs Similarity: Confusing $\frac{AD}{DB} = \frac{AE}{EC}$ (BPT) with $\frac{AD}{AB} = \frac{DE}{BC}$ (Similarity).

Drawing Figures: Not drawing a figure for geometry questions. It is mandatory for marks.

Scoring Tips 🏆

Correspondence: Always check which angle corresponds to which before writing similarity.

Given/To Prove: Always write "Given", "To Prove", and "Construction" headings in theorem proofs.

Shadow Problems: Remember "at the same time" means the angle of elevation of the sun is equal for both objects.

Self-Assessment Mock Test (10 Marks)

Q1 (1M): All equilateral triangles are ___________ (congruent/similar).

Q2 (2M): In $\triangle PQR$, $S$ and $T$ are points on $PQ$ and $PR$ such that $ST || QR$. If $PS=3$, $SQ=3$, $PT=4$, find $PR$.

Q3 (3M): ABCD is a trapezium with $AB || DC$. Diagonals intersect at O. If $AB = 2CD$, find ratio of areas of $\triangle AOB$ and $\triangle COD$.

Q4 (4M): Prove the Basic Proportionality Theorem.