Question

class 10
prove the BPT theorm​

Answer 1

EXPLANATION.

Basic proportionality Theorem.

As we know that,

Theorem = If a line is drawn parallel on one sides of a triangle to intersect the other two sides in distinct points, the other two sides are divide in the same ratio.

⇒ In ΔABC.

⇒ PQ ║BC.

To prove = AP/PB = AQ/QC.

Constructions = join BE and CD.

Draw PM ⊥ AC & QN ⊥ AB.

⇒ In ΔAPQ.

Area of triangle(Δ) = 1/2 x base x height.

⇒ 1/2 x AP x QN ⇒(1).

⇒ In ΔAPQ.

Area of triangle(Δ) = 1/2 x base x height.

⇒ 1/2 x AQ x PM ⇒(2).

⇒ In ΔBPQ.

Area of triangle(Δ) = 1/2 x base x height.

⇒ 1/2 x PB x QN ⇒(3).

⇒ In ΔPQC.

Area of triangle(Δ) = 1/2 x base x height.

⇒ 1/2 x QC x PM ⇒(4).

As we know that,

Divide equation (1) & (3), we get.

⇒ (ΔAPQ)/(ΔBPQ) = 1/2 x AP x QN/1/2 x PB x QN.

⇒ (ΔAPQ)/(ΔBPQ) = AP/PB.

Divide equation (2) & (4), we get.

⇒ (ΔAPQ)/(ΔPQC) = 1/2 x AQ x PM/1/2 x QC x PM.

⇒ (ΔAPQ)/(ΔPQC) = AQ/QC.

We get,

⇒ (ΔAPQ)/(ΔBPQ) = (ΔAPQ)/(ΔPQC).

⇒ AP/PB = AQ/QC.

HENCE PROVED.