Triangles on the Same Base & between Same Parallels are Equal in Area


Here we will prove that triangles
on the same base and between the same parallels are equal in area.

Given: PQR and SQR are two triangles on the same base QR and
are between the same parallel lines QR and MN, i.e., P and S are on MN.

To prove: ar(∆PQR) = ar(∆SQR).

Construction: Draw QM RP cutting MN at M.


Proof:

            Statement

            Reason

1. QRPM is a parallelogram.

1. MP ∥ QR and QM ∥ RP by construction.

2. ar(∆PQR) = (frac{1}{2}) × ar(parallelogram QRPM).

ar(∆SPQ) = (frac{1}{2}) × ar(parallelogram QRPM).

2. Area of a triangle = (frac{1}{2}) × area of a parallelogram, on the same base, and between the same parallels.

3. ar(∆PQR) = ar(∆SQR). (Proved)

3. From statements in 2.

Corollaries:

(i) Triangles with equal bases and between the same parallels
are equal in area.

(ii) If two triangles have equal bases, ratio of their areas =
ratio of their altitudes.

(iii) If two triangles have equal altitudes, ratio of their
areas = ratio of their bases.

(iv) A median of a triangle divides the triangle in two
triangles of equal area.

9th Grade Math

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