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प्रश्न
In the given figure, XY || AC and XY divides ΔABC into two regions, equal in area. Show that `(AX)/(AB) = ((2 - sqrt(2)))/2`.
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उत्तर
Given:
In triangle ABC, XY || AC, with X on AB and Y on BC.
The segment XY divides ΔABC into two regions of equal area.
Show `(AX)/(AB) = ((2 - sqrt(2)))/2`.
Step-wise calculation:
1. Because XY || AC, triangles BXY and BAC are similar (corresponding vertices B→B, X→A, Y→C).
2. Let AB = 1 (work with a unit length along AB); let BX = t, so AX = 1 – t.
3. Similarity gives `(Area(ΔBXY))/(Area(ΔBAC)) = ((BX)/(BA))^2 = t^2`.
4. The condition “XY divides ΔABC into two equal areas”
⇒ Area(ΔBXY) = `1/2` × Area(ΔBAC).
Therefore `t^2 = 1/2`.
5. So `t = 1/sqrt(2)`.
Hence `(AX)/(AB) = 1 - t`
= `1 - 1/sqrt(2)`
6. Simplify `1 - 1/sqrt(2)`:
`1 - 1/sqrt(2) = sqrt(2)/sqrt(2) - 1/sqrt(2)`
= `(sqrt(2) - 1)/sqrt(2)`
= `(2 - sqrt(2))/2`
`(AX)/(AB) = ((2 - sqrt(2)))/2`. This is the required result.
