For each of the following statements state whether true (T) or false (F):
(i) Two circles with different radii are similar.
(ii) Any two rectangles are similar.
(iii) If two triangles are similar then their corresponding angles are equal and their corresponding sides are equal.
(iv) The length of the line segment joining the midpoints of any two sides of a triangle is equal to half the length of the third side.
(v) In a ΔABC, AB = 6 cm, ∠A = 45° and AC = 8 cm and in a ΔDEF, DF = 9 cm, ∠D = 45° and DE = 12 cm, then ΔABC ~ ΔDEF.
(vi) The polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus.
(vii) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding angle-bisector segments.
(viii) The ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding medians.
(ix) If O is any point inside a rectangle ABCD then OA2 + OC2 = OB2 + OD2.
(x) The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.
(i) T
Two similar figures have the same shape but not necessarily the same size. Therefore, all circles are similar.
(ii) F
Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion).
Consider an example,
Let a rectangle have sides 2cm and 3cm and another rectangle have sides 2cm and 5cm.
Here, the corresponding angles are equal but the corresponding sides are not in the same ratio.
(iii) F
Two triangles are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in the same ratio (or proportion).
(iv) T
Midpoint Theorem states that the line joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
(v) F
Two triangles are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in the same ratio (or proportion).
But here, the corresponding sides are
AB/DE = 6/12 = 1/2 and AC/DF = 8/9
AB/DE ≠ AC/DF
(vi) F
The polygon formed by joining the midpoints of sides of any quadrilateral is a parallelogram.
(vii) T
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
(viii) T
The perimeters of the two triangles are in the same ratio as the sides. The corresponding medians also will be in this same ratio.
(ix) T
Let us construct perpendiculars OP, OQ, OR and OS from point O.
Let us take LHS = OA2 + OC2
From Pythagoras theorem,
= (AS2 + OS2) + (OQ2 + QC2)
As also AS = BQ, QC = DS and OQ = BP = OS,
= (BQ2 + OQ2) + (OS2 + DC2)
Again by Pythagoras theorem,
= OB2 + OD2 = RHS
As LHS = RHS, hence proved.
(x) T
In rhombus ABCD, AB = BC = CD = DA.
We know that diagonals of a rhombus bisect each other perpendicularly.
i.e. AC ⊥ BD, ∠AOB = ∠BOC = ∠COD = ∠AOD = 90° and
OA = OC = AC/2, OB = OD = BD/2
Let us consider right angled triangle AOB.
By Pythagoras theorem,
AB2 = OA2 + OB2
⇒ AB2 = (AC/2)2 + (BD/2)2
⇒ AB2 = AC2/4 + BD2/4
⇒ 4AB2 = AC2+ BD2
⇒ AB2 + AB2 + AB2 + AB2 = AC2+ BD2
∴ AB2 + BC2 + CD2 + DA2 = AC2+ BD2