Show that the areas of the shaded blue triangle and the shaded red triangle are equal. Class 9

Show that the areas of the shaded blue triangle and the shaded red triangle are equal. Class 9

Question 1.

Show that the areas of the shaded blue triangle and the shaded red triangle are equal. Class 9

Find a way of cutting up the blue triangle into some number of pieces and rearranging the pieces to cover the red triangle.

Solution:

Given:

A triangle in which a vertex is joined to the points of trisection of the opposite side.

To prove:

Blue triangle area = Red triangle area

Let point D and E be the points of trisection of ∆ABC.

∵ BD = DE = EC ...(i)

∵ Area of triangle = $\frac{1}{2}$ × base × height

A(∆ABD) = $\frac{1}{2}$ × BD × height

A(∆ADE) = $\frac{1}{2}$ × DE × height

A(∆AEC) = $\frac{1}{2}$ × EC × height

∵ ∆ABD, ∆ADE and ∆ have same vertex A and lie on same line BC

∵ Altitude from A is same ... (v)

∵ ∆ABD = ∆ADE = ∆AEC ...[From (i), (ii), (iii), (iv)]

Way of cutting blue triangle:

Red and Blue triangles are congruent but they are different in shape. So one approach is to cut the blue triangle along a line parallel to a suitable side and rearrange the pieces to match the red triangle shape.