प्र.4 पुढील प्रश्न सोडवा. (कोणतेही दोन)
1) रेषा PQहीABC च्या बाजू AB बाजू AC
ला अनुक्रमे P व Q मध्ये छेदते तर सिध्द करा,
A/AAPQ)-AP AQ
AAABC) ABXAC
Answers
Given : PQ join AB and AC at mid point
रेषा PQहीABC च्या बाजू AB बाजू AC ला अनुक्रमे P व Q मध्ये छेदते
To Find : Prove that
Ar Δ APQ / Ar ΔABC = AP * A Q / AB * AC
Solution:
line joining the mid-point of two sides of a triangle is equal to half the length of the third side and parallel to third side
PQ || BC
ΔAPQ and ΔABC
∠A = ∠A common
∠P = ∠B (corresponding angles )
∠Q = ∠C (corresponding angles )
=> ΔAPQ ≈ ΔABC
AP/AB = AQ/AC = PQ/BC
Ratio of area of similar triangles = square of ratio of corresponding sides
Ar ΔAPQ / Ar ΔABC = ( AP/ AB)²
=> Ar ΔAPQ / Ar ΔABC = ( AP/ AB) ( AP/ AB)
AP/AB = AQ/AC
=> Ar ΔAPQ / Ar ΔABC = ( AP/ AB) ( AQ/ AC)
=> Ar Δ APQ / Ar ΔABC = AP * A Q / AB * AC
QED
Hence proved
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