In this chapter, you will learn
- —Understand similarity criteria for triangles (AA, SSS, SAS)
- —Apply Basic Proportionality Theorem in solving problems
- —Prove and apply Pythagoras theorem
- —Calculate areas of similar triangles
- —Solve problems on angle bisectors and medians
- —Apply similar triangles in real-world contexts
Similarity Criteria
Similar Triangles: Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
Three Similarity Criteria:
1. AA (Angle-Angle) Criterion: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
2. SSS (Side-Side-Side) Criterion: If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.
3. SAS (Side-Angle-Side) 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 triangles are similar.
Example: AA Similarity
In triangles ABC and DEF, ∠A = ∠D and ∠B = ∠E, then triangle ABC ~ triangle DEF
This means: AB/DE = BC/EF = CA/FD
Properties of Similar Triangles:
- Ratio of corresponding sides is constant
- Ratio of areas = (Ratio of corresponding sides)²
- Ratio of perimeters = Ratio of corresponding sides
- Corresponding angles are equal
Exam Tip
Similarity criteria are frequently asked in board exams. Always identify which criterion applies (AA, SSS, or SAS) and write the similarity statement correctly using the tilde (~) symbol.
Common Mistake
Students often confuse congruence with similarity. Remember: congruent triangles are equal in size and shape, but similar triangles only have the same shape (sides proportional, angles equal).
Basic Proportionality Theorem (Thales' Theorem)
Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Statement: In triangle ABC, if DE is parallel to BC (where D is on AB and E is on AC), then:
AD/DB = AE/EC
or AD/AB = AE/AC
Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Example:
In triangle ABC, DE || BC. If AD = 3 cm, DB = 6 cm, AE = 2 cm, find EC.
Using Basic Proportionality Theorem:
AD/DB = AE/EC
3/6 = 2/EC
EC = 4 cm
Key Points:
- The parallel line divides the sides in the same ratio
- The theorem can be stated in multiple forms
- Used to find unknown lengths in triangles
- Basis for constructing parallel lines
Exam Tip
BPT is often used in conjunction with similar triangles. Practice problems involving finding unknown lengths when a line is parallel to one side.
Common Mistake
Students forget to set up the proportion correctly. Remember: AD/DB = AE/EC (same side divided by each other). Don't mix up the ratios.
Pythagoras Theorem and Similar Triangles
Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
Pythagoras Theorem:
In right triangle ABC with right angle at C:
AB² = AC² + BC²
or c² = a² + b²
where c is the hypotenuse, a and b are the other sides
Converse of Pythagoras Theorem: If in a triangle, the square of one side equals the sum of squares of the other two sides, then the triangle is right-angled.
Example: Using Pythagoras Theorem
In a right triangle, two sides are 3 cm and 4 cm. Find the hypotenuse.
c² = 3² + 4² = 9 + 16 = 25
c = 5 cm
Relationship with Similar Triangles:
- If altitude is drawn to hypotenuse, three similar triangles are formed
- The altitude is geometric mean of segments of hypotenuse
- Each leg is geometric mean of hypotenuse and adjacent segment
Common Pythagorean Triplets:
- (3, 4, 5) - multiply by 2, 3, 4... to get (6, 8, 10), (9, 12, 15), (12, 16, 20)
- (5, 12, 13)
- (8, 15, 17)
- (7, 24, 25)
Exam Tip
Pythagoras theorem is one of the most important theorems. Learn the triplets as shortcuts. It's frequently used in 3-5 mark questions.
Common Mistake
Students sometimes forget to square all terms. c² = a² + b² means the SQUARE of hypotenuse, not just c = a + b. Also, identify which is the hypotenuse (longest side opposite right angle).