Area of Triangles

In today's blog, I present some very elementary proofs regarding area. This is needed by the proofs on similar triangles which I use as background for sin and cosin.

I present these definitions and proofs more for a sense of completeness.

Definition 1: Rectangle

A rectangle is a parallelogram where all angles are 90 degrees.

Definition 2: Area of a rectangle

The area of a rectangle is width * height.

Definition 3: Right Triangle

A right triangle is triangle where one of its angles is 90 degrees.

Lemma 1: The area of a parallelogram is base * height





















Proof:

This follows directly from Lemma 2, here since we can construct a rectangle in the same parallel based on the base of the parallelogram.

By Lemma 2, the parallelogram will be congruent to this rectangle so the area of the parallelogram will be the same.

QED

Lemma 2: The area of any triangle is (1/2)height * base

















Proof:

(1) Let ABC be a triangle

(2) Let CE be a line parallel to AB

(3) Let AF be a line parallel to BC

(4) let D be the point where CE and AF intersect.

(5) From (2) and (3), we see that ABCD is a parallelogram. [See here for definition of a parallelogram]

(6) Then triangle ABC ≅ triangle CDA by S-A-S since [See here for definition of S-A-S]:

(a) AB ≅ CD and BC ≅ DA since opposite sides of a parallelogram are congruent [See here for proof]

(b) ∠ ABC ≅ ∠ CDA since opposite angles of a parallelogram are congruent [See here for proof]

(7) Now since the area of the parallelogram is itself is base*height (see Lemma 1 above), the area of each triangle is (1/2)base*height.

QED

References

• Euclid's Elements, David E. Joyce, Professor of Mathematics and Computer Science, Clarks University