Question

The lengths of the

sides of triangle ABC
are a, b, c
centimeters. The
vertices of the
triangle DEF are the
midpoints of the
sides ABC of the
triangle.
Which statement is
incorrect?

Answer 1

`S_(\Delta ABC)` = 4·
`S_(\Delta EDC)`

Explanation:

From the midpoint theorem, which states that the line that a line drawn such that it joins the midpoints of two sides of a triangle, is parallel to the third side of the triangle and is equal to half the length of the third side

Therefore, the lengths of the sides of ΔDEF, drawn by joining the midpoints of ΔABC is equal to half the length of and parallel to the corresponding side of ΔABC

We therefore, have that the corresponding sides of ΔABC and ΔDEF have a common ratio and a pair of sides in each triangle form same angles, therefore;

ΔDEF is similar to ΔABC by Side, Side, Side SSS similarity.

The length of the perimeter of ΔABC,
`S_(\Delta ABC)` = 2 × The length of the perimeter of triangle ΔEDC,
`S_(\Delta EDC)`

`S_(\Delta ABC)` = 2 ×
`S_(\Delta EDC)`

∴
`S_(\Delta ABC)` ≠ 4 ×
`S_(\Delta EDC)`

The statement which is incorrect is therefore;

`S_(\Delta ABC)` = 4 ×
`S_(\Delta EDC)`.