Class 10 — Mathematics (NCERT)

Triangles: Similarity Criteria, Areas of Similar Triangles, and Pythagoras Theorem

Class 10

✓Understand and apply the Angle-Angle (AA), Side-Side-Side (SSS), and Side-Angle-Side (SAS) similarity criteria for triangles.
✓State and apply the theorem relating the ratio of the areas of two similar triangles to the ratio of the squares of their corresponding sides.
✓State, prove, and apply the Pythagoras Theorem and its converse to solve problems involving right-angled triangles.
✓Solve problems involving a combination of similarity and Pythagoras Theorem.

Key concepts

Similarity Criteria for Triangles

Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional).\n\n1. **AA (Angle-Angle) Similarity Criterion**: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.\n2. **SSS (Side-Side-Side) Similarity Criterion**: If the corresponding sides of two triangles are proportional, then their corresponding angles are equal and hence the two triangles are similar.\n3. **SAS (Side-Angle-Side) Similarity Criterion**: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

Theorem on Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. If ΔABC ~ ΔPQR, then the ratio of their areas is given by the square of the ratio of their corresponding sides.

Area(ΔABC) / Area(ΔPQR) = (AB/PQ)² = (BC/QR)² = (CA/RP)²

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). If ΔABC is a right-angled triangle with the right angle at B, then AC² = AB² + BC².

Hypotenuse² = Perpendicular² + Base² (or AC² = AB² + BC²)

Converse of Pythagoras Theorem

If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. For example, if in ΔABC, AC² = AB² + BC², then ∠B = 90°.

Key facts to remember

1Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
2AA Similarity Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
3SSS Similarity Criterion: If the corresponding sides of two triangles are proportional, the triangles are similar.
4SAS Similarity Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.
5Theorem on Areas of Similar Triangles: The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
6Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
7Converse of Pythagoras Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
8The ratio of corresponding altitudes, medians, and perimeters of similar triangles is equal to the ratio of their corresponding sides.

Worked examples

Example 1

In the given figure, if PQ || RS, prove that ΔPOQ ~ ΔSOR. (Assume a figure where lines PR and QS intersect at O, forming triangles POQ and SOR, with PQ parallel to RS.)

IGiven: PQ || RS.

IIConsider ΔPOQ and ΔSOR.

III∠POQ = ∠SOR (Vertically opposite angles).

IV∠OPQ = ∠OSR (Alternate interior angles, since PQ || RS and PS is a transversal).

V∠PQO = ∠SRO (Alternate interior angles, since PQ || RS and QR is a transversal).

VITherefore, by AAA similarity criterion (or AA similarity criterion, as two angles are sufficient), ΔPOQ ~ ΔSOR.

Answer

Hence proved.

The AA similarity criterion is sufficient to prove similarity, as the third angle will automatically be equal.

Example 2

ΔABC ~ ΔDEF. If Area(ΔABC) = 64 cm² and Area(ΔDEF) = 121 cm² and EF = 15.4 cm, find BC.

IGiven: ΔABC ~ ΔDEF.

IIArea(ΔABC) = 64 cm², Area(ΔDEF) = 121 cm², EF = 15.4 cm.

IIIWe know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

IVArea(ΔABC) / Area(ΔDEF) = (BC/EF)²

V64 / 121 = (BC / 15.4)²

VITaking square root on both sides: √(64/121) = BC / 15.4

VII8 / 11 = BC / 15.4

VIIIBC = (8 * 15.4) / 11

9BC = 8 * 1.4

10BC = 11.2 cm.

Answer

BC = 11.2 cm.

Remember to take the square root of the area ratio before equating it to the side ratio.

Example 3

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.

ILet the ladder be AC, the window be at point A, and the base of the wall be B. The wall is perpendicular to the ground, forming a right-angled triangle ΔABC at B.

IIGiven: Length of ladder (hypotenuse AC) = 10 m.

IIIHeight of window from ground (perpendicular AB) = 8 m.

IVLet the distance of the foot of the ladder from the base of the wall (base BC) be x m.

VBy Pythagoras Theorem, AC² = AB² + BC².

VI10² = 8² + x²

VII100 = 64 + x²

VIIIx² = 100 - 64

9x² = 36

10x = √36

11x = 6 m (Distance cannot be negative).

Answer

The distance of the foot of the ladder from the base of the wall is 6 m.

Always draw a diagram to visualise the problem and label the sides correctly.

Common mistakes

✗Confusing similarity with congruence; similar triangles have proportional sides, while congruent triangles have equal sides.
✗Incorrectly identifying corresponding sides or angles when setting up ratios for similar triangles, leading to incorrect calculations.
✗Forgetting to square the ratio of sides when calculating the ratio of areas of similar triangles, or incorrectly taking the square root when finding side ratios from area ratios.
✗Applying the Pythagoras Theorem to triangles that are not right-angled, or not correctly identifying the hypotenuse.
✗Errors in algebraic manipulation, especially when dealing with squares and square roots.

Exam tips

★Always draw a neat and labelled diagram for geometry problems to visualise the situation and identify given information clearly.
★Clearly state the similarity criterion (AA, SSS, or SAS) used when proving triangles similar in your solution.
★For theorems, understand the proof thoroughly, as direct proofs of theorems (like Pythagoras Theorem or the Areas of Similar Triangles Theorem) are often asked in exams.
★Ensure all steps are shown clearly and logically, especially for proofs and multi-step problems, to gain full marks.
★Double-check all calculations, particularly when dealing with squares and square roots in problems involving areas and Pythagoras Theorem.

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