Triangles: Core Theorems - Class 10 Boards Mathematics

Triangles: Core Theorems and Similarity

Basic Proportionality Theorem (BPT) / Thales Theorem

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those sides in the same ratio. This theorem is fundamental for solving many triangle problems and is frequently asked in exams.

• If $DE \parallel BC$ in triangle $ABC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.
• The theorem is also known as Thales Theorem.
• The proof involves constructing perpendiculars and using the area formula for triangles multiple times.

Proof of BPT…

Triangles: Core Theorems and Similarity

Basic Proportionality Theorem (BPT) / Thales Theorem

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those sides in the same ratio. This theorem is fundamental for solving many triangle problems and is frequently asked in exams.

• If $DE \parallel BC$ in triangle $ABC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.
• The theorem is also known as Thales Theorem.
• The proof involves constructing perpendiculars and using the area formula for triangles multiple times.

Proof of BPT

The proof of BPT is exam-critical and involves using triangle areas and perpendiculars. It is repeatedly asked in board exams.

• Construct perpendiculars $DM \perp AC$ and $EN \perp AB$ from points $D$ and $E$.
• Join $BE$ and $CD$.
• Use area formulas: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ for triangles $ADE$ and $BDE$.
• Divide the area equations to get $\frac{AD}{DB} = \frac{AE}{EC}$.
• The proof relies on the fact that triangles on the same base and between the same parallels have equal areas.

Converse of BPT

The converse states that if a line divides two sides of a triangle in the same ratio, then it is parallel to the third side. This is also frequently tested.

• If $\frac{AD}{DB} = \frac{AE}{EC}$, then $DE \parallel BC$.
• The proof involves assuming $DE$ is not parallel, constructing another parallel, and showing the ratios force $E$ and $F$ to coincide, thus $DE \parallel BC$.

Types of BPT Applications

BPT can be applied in three forms depending on the question's requirements. Recognizing which ratio to use is essential for solving problems efficiently.

• Small/Small Ratio: $\frac{AD}{DB} = \frac{AE}{EC}$
• Small/Big Ratio (Upper): $\frac{AD}{AB} = \frac{AE}{AC}$
• Small/Big Ratio (Lower): $\frac{DB}{AB} = \frac{EC}{AC}$
• The choice depends on which sides are given and what needs to be proved.

Solving BPT-Based Questions

Questions often provide parallel lines and ask for unknown side lengths or ratios. The method is to set up the appropriate ratio and solve for the variable.

• If $DE \parallel BC$, and $AD = x$, $DB = x-2$, $AE = x+2$, $EC = x-1$, set up $\frac{x}{x-2} = \frac{x+2}{x-1}$ and solve for $x$.
• For checking parallelism, compare ratios: if $\frac{AX}{XB} \neq \frac{AY}{YC}$, then $XY$ is not parallel to $BC$.

Similarity of Triangles

Similarity means triangles have the same shape, but not necessarily the same size. All corresponding angles are equal, and corresponding sides are in proportion.

• If $\triangle ABC \sim \triangle PQR$, then $\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$.
• The ratios of corresponding sides are equal: $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$.
• Congruent triangles are always similar, but similar triangles are not always congruent.

Criteria for Similarity

There are three main criteria for proving triangles are similar. Each criterion is based on specific equalities of angles and/or side ratios.

• AA (Angle-Angle): If two angles of one triangle are equal to two angles of another, triangles are similar.
• SAS (Side-Angle-Side): If two sides are in proportion and the included angle is equal, triangles are similar.
• SSS (Side-Side-Side): If all three sides are in proportion, triangles are similar.
• The pattern of correspondence must be maintained when writing similarity.

Corresponding Parts of Similar Triangles (CPST)

Once triangles are proven similar, all corresponding parts (angles and sides) are equal or proportional. This is used to deduce further relationships.

• If triangles are similar, use CPST to write ratios and equalities for sides and angles.
• For example, if $\triangle CBA \sim \triangle QPR$, then $\frac{CB}{QP} = \frac{BA}{PR} = \frac{CA}{QR}$.

Special Properties and Cases

Certain shapes and segments always exhibit similarity. Recognizing these helps in quick problem-solving.

• Any two line segments are always similar.
• Any two circles are always similar.
• Any two squares are always similar.
• Any two equilateral triangles are always similar.
• For polygons other than triangles, both angles and side ratios must be equal for similarity.

Ratio Properties in Similar Triangles

In similar triangles, ratios extend beyond sides to medians, altitudes, angle bisectors, and perimeters.

• Ratio of corresponding medians, altitudes, and angle bisectors equals the ratio of corresponding sides.
• Ratio of perimeters equals the ratio of corresponding sides.
• If side ratio is $1/2$, perimeter ratio is also $1/2$.

Area Ratio in Similar Triangles

The ratio of areas of similar triangles is the square of the ratio of corresponding sides.

• If side ratio is $a:b$, then area ratio is $a^2:b^2$.
• For example, if side ratio is $3:5$, area ratio is $9:25$.

Exam-Focused Takeaways

Mastering BPT, its converse, and similarity criteria is essential for solving triangle problems in exams. Recognize which theorem or criterion applies, maintain correct correspondence, and use CPST for deducing unknowns.

• Always check for parallel lines to apply BPT.
• For similarity, look for angle equalities or side ratios.
• Use medians, altitudes, and angle bisectors ratios only if triangles are similar.
• Area and perimeter ratios follow side ratios in similar triangles.
• Practice constructing diagrams and identifying which triangles to compare for similarity or proportionality.

Word count: 1197