A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas. Class 8

A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas. Class 8

Question 1.

A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas. Class 8

Solution:

Join the centre O with other vertices as shown.

Here, ADEF is a trapezium made of 3 congruent equilateral triangles, BCDO is a rhombus made of 2 congruent equilateral triangles and ABO is an equilateral triangle.

Clearly ratio of equilateral triangle, rhombus and trapezium (as per areas) is 1 : 2 : 3.

Question 2.

ZYXW is a trapezium with ZY || WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of ∆ZWB. Class 8

Solution:

In ∆AYZ and ∆AXB,

∠ZAY = ∠BAX (Vertically opposite angles)

AY = AX (Given)

∠Y = ∠X (Alternate angles)

So, ∆AYZ ≅ ∆AXB (By ASA)

So, area of ∆AYZ = area of ∆AXB ...(1)

Now, area of trapezium ZYXW

= Area of ZAXW + Area of ∆ZYA

= Area of ZAXW + Area of ∆AXB

= Area of ∆ZWB (Shown)