Verify that the area of the triangle with vertices (2, 3), (5, 7) and

1.	Verify that the area of the triangle with vertices (2, 3), (5, 7) and (-3 -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).
Answer» Given: The points (2, 3), (5, 7), and (-3, -1). The area of triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is = 1/2 [x₁(y₂ – y₃) + x₂(y₃ - y₁) + x₃(y₁ – y₂)] The area of given triangle = 1/2 [2(7 + 1) + 5(-1 - 3) – 3(3 - 7)] = 1/2 [16 – 20 + 12] = 1/2 [8] = 4 Origin shifted to point (-1, 3), the new coordinates of the triangle are (3, 0), (6, 4), and (-2, -4) obtained from subtracting a point (-1, 3). The new area of triangle = 1/2 [3(4 - (-4)) + 6(-4 - 0) – 2(0 - 4)] = 1/2 [24 - 24 + 8] = 1/2 [8] = 4 Since the area of the triangle before and after the translation after shifting of origin remains same, i.e. 4. ∴ We can say that the area of a triangle is invariant to shifting of origin.

Verify that the area of the triangle with vertices (2, 3), (5, 7) and (-3 -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).

Answer»

Given:

The points (2, 3), (5, 7), and (-3, -1).

The area of triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is

= 1/2 [x₁(y₂ – y₃) + x₂(y₃ - y₁) + x₃(y₁ – y₂)]

The area of given triangle = 1/2 [2(7 + 1) + 5(-1 - 3) – 3(3 - 7)]

= 1/2 [16 – 20 + 12]

= 1/2 [8]

= 4

Origin shifted to point (-1, 3), the new coordinates of the triangle are (3, 0), (6, 4), and (-2, -4) obtained from subtracting a point (-1, 3).

The new area of triangle = 1/2 [3(4 - (-4)) + 6(-4 - 0) – 2(0 - 4)]

= 1/2 [24 - 24 + 8]

= 1/2 [8]

= 4

Since the area of the triangle before and after the translation after shifting of origin remains same, i.e. 4.

∴ We can say that the area of a triangle is invariant to shifting of origin.

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