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  • The mid-points of the sides of a triangle are (5, 7, 11), (0, 8, 5) and (2, 3, – 1). Find its vertices.

The mid-points of the sides of a triangle are (5, 7, 11), (0, 8, 5) and (2, 3, – 1). Find its vertices.

Answers (1)

Given:

Mid-points of sides of triangle ABC are,

D(5,7,11), E(0,8,5) & F(2,3,-1)

Let us assume that the vertices of the triangle are –

A(x1, y1, z1), B(x2, y2, z2) & C(x3, y3, z3).

Now, we know that E is the mid-point of AC, thus,

(x1 + x3/2, y1 + y3/2, z1 + z3/2) = (0,8,5)

Thus,

C(x3, y3, z3) = C(-x1, 16 – y1, -2-z­1)

Now, mid-point of AB, F is

(x1 + x2/2, y1 + y2/2, z1 + z2/2) = (2,3,-1)

Now, B(x2, y2, z2) = B(4-x1, 6-y1, -1-z1)

Now, mid-point of BC, D is,

-x1 + 4 – x1/2 = 5,

16 – y1 + 6 – y1/2 = 7

& 10 – z1 – 2 – z1/2 = 11

Thus, x1 = -3, y1 = 4 & z1 = -7

Thus, A = (-3,4,-7), B = (7,2,5) & C = (3,12,17)

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