Triangles
This chapter covers similar figures and similar triangles, the criteria for similarity of triangles (AAA, SSS, SAS), and the Pythagoras theorem and its converse.
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional). Similar triangles have the same shape but may differ in size.
If a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, the other two sides are divided in the same ratio. This is one of the most frequently used theorems for proving similarity and solving problems.
Q. State the Pythagoras theorem and its converse.
Pythagoras theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (the legs). Its converse: if the square of one side of a triangle equals the sum of squares of the other two sides, then the angle opposite the first side is a right angle.
Criteria for similarity of triangles: AAA (or AA) - all corresponding angles equal; SSS - all corresponding sides proportional; SAS - one pair of corresponding sides proportional and the included angles equal.
Q. The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find the length of each side of the rhombus.
The diagonals of a rhombus bisect each other at right angles, so each side forms the hypotenuse of a right triangle with legs 12 cm and 5 cm (half of each diagonal). By Pythagoras theorem: side = root(12^2 + 5^2) = root(144 + 25) = root(169) = 13 cm.
If triangle ABC is similar to triangle DEF, and AB/DE = 2/3, what is the ratio of their areas?
Correct answer: (B) 4:9
The ratio of the areas of two similar triangles is equal to the _______ of the ratio of their corresponding sides.
Square
All congruent triangles are similar, but all similar triangles are not congruent.
True — congruent triangles are identical in shape and size, so they satisfy the similarity condition too, but similar triangles only need the same shape (proportional sides), not necessarily the same size, so they need not be congruent.