If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that QPQGQP¯=3QG¯. - Mathematics and Statistics

प्रश्न

If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that `bar"QP" = 3bar"QG"`.

योग

उत्तर

Let `bar"p" and bar"g"` be the position vectors of P and G w.r.t. the circumcentre Q.

i.e. `bar"QR" = bar"p" and bar"QG" = bar"g"`

We know that Q, G, P are collinear and G divides segment QP internally in the ratio 1 : 2.

∴ by section formula for internal division,

`bar"g" = (1.bar"p" + 2bar"q")/(1 + 2) = bar"p"/3` .....`[∵ bar"q" = 0]`

∴ `bar"p" = 3bar"g"`

∴ `bar"QP" = 3bar"QG".`

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Vectors and Their Types

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q II. 19)Q II. 18)Q II. 20)

अध्याय 5: Vectors - Miscellaneous exercise 5 [पृष्ठ १९१]

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बालभारती Mathematics and Statistics 1 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board

अध्याय 5 Vectors
Miscellaneous exercise 5 | Q II. 19) | पृष्ठ १९१

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