In triangle PQR ,if QS is angle bisector of angle Q , then show that A( triangle PQS) upon A( triangle QRS) is equal to PQ upon QR

Question

In triangle PQR ,if QS is angle bisector of angle Q , then show that A( triangle PQS) upon A( triangle QRS) is equal to PQ upon QR

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Kylie 2 months 2021-11-28T11:51:03+00:00 1 Answer 0 views 0

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    2021-11-28T11:52:37+00:00

    Step-by-step explanation:

    in Δ pqr if qs bisect ∠q then Area of Triangle sqr / Area of Triangle pqs = qr/pq

    Step-by-step explanation:

    Correct Question : A(triangle sqr)/A(triangle pqs)= qr/pq

    Let say Angle q = 2β

    Then qs is bisector of ∠q

    => ∠pqs = ∠rqs = β

    now Draw sm ⊥ pq & sn ⊥ qr

    Area of Triangle pqs

    = (1/2) * pq * sm

    in Δ smq

    sinβ = sm/qs

    => sm = qs * sinβ

    Area of Triangle pqs = (1/2) * pq *qs * sinβ

    Area of Triangle sqr

    = (1/2) * qr * sn

    in Δ snq

    sinβ = sn/qs

    => sn = qs * sinβ

    Area of Triangle sqr = (1/2) * qr *qs * sinβ

    Area of Triangle sqr / Area of Triangle pqs = (1/2) * qr *qs * sinβ / (1/2) * pq *qs * sinβ

    => Area of Triangle sqr / Area of Triangle pqs = qr/pq

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