Question

Which all statements are true :

(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 $\Delta ABC,AB=6cm,\angle A={45}^{o}andAC=8cm$ and in a $\Delta DEF$ DF = 9 cm, $\angle D={45}^o$ and DE = 12 cm, then $\Delta ABC\sim \Delta DEF$.

(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 $O{A}^2+O{C}^2=O{B}^2+O{D}^2$.

(x) The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.

Answer 1

(i) True

(ii) True

(iii) True

(iv) True

(v) True.

$\frac{AB}{DF}=\frac{6}{9}=\frac{2}{3}\frac{AC}{DE}=\frac{8}{12}=\frac{2}{3}\frac{AB}{DF}=\frac{AC}{DE}\angle D=\angle A={45}^o\therefore \Delta ABC\sim \Delta DEF$

(vi) False. The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are perpendicular is a rectangle. The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are congruent is a rhombus.

(vii) True.

(viii) True.

(ix) True.

(x) True.