If  ∆ ABC ~  ∆RPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10 cm, then find QR

State and prove Converse of Pythagoras’ Theorem

If in a right angle ABC, right angled at A, AD ⊥ BC, then prove that AB² + CD² = BD² + AC²

Prove that the sum of the square of the sides of a rhombus is equal to the sum of squares of its diagonals

In ∆ABC, <B = 90°, BD ⊥ AC, area (∆ABC) = A and BC = a, then prove that

If one diagonal of a trapezium divides the other diagonal in the ratio 1 : 3. Prove that one of the parallel sides is three times the other.

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite the first side is a right angle.
Using the above, do the following:
In an isosceles triangle PQR, PQ = QR and PR² = 2PQ². . Prove that <Q is a right angle.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.
Using the above, do the following:
In figure, BA || QR, and CA || SR, prove that the other two sides are divided in the same ratio.

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Using the above, do the following:
AD is an altitude of an equilateral triangle ABC. On AD as the base, another equilateral triangle ADE is constructed. Prove that, area (∆ADE): area (∆ABC) =3:4

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.
Using the above, do the following:
In figure, PQ || AB and AQ || CB.
Prove that AR² = PR. CR.

If the areas of two similar triangles are in ratio 25 : 64, write the ratio of their corresponding sides.

in a right angle triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. ‘
Using the above, do the following:
Prove that in a ∆ABC, if AD is perpendicular to BC, then AB² + CD² = AC² + BD².

The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Using the above, prove the following: The areas of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described, on its diagonal.