In $Δ A B C, A P ⊥ B C a n d B Q ⊥ A C$ which intersect each other at $O$ . Prove that $A O \times O P = B O \times O Q$ .

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Solution

$G i v e n : I n △ A B C, A P ⊥ B C a n d B Q ⊥ A C w h i c h i n t e r s e c t e a c h o t h e r a t O . T o P r o v e : A O \times O P = O B \times O Q P r o o f : I n △ A B C w e h a v e W e h a v e, ∠ A P B = 90 a n d ∠ A Q B = 90 [∵ A P ⊥ B C a n d B Q ⊥ A C] I n △ A O Q a n d △ B O P ∠ A O Q = ∠ B O P (V e r t i c a l l y o p p o s i t e a n g l e s) ∠ O Q A = ∠ O P B = 90 H e n c e t r i a n g l e s a r e s i m i l a r b y A A s i m i l a r i t y △ A O Q \sim △ B O P \Rightarrow \frac{A O}{O B} = \frac{O Q}{O P} \Rightarrow A O \times O P = O B \times O Q H e n c e P r o v e d$

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In ΔABC,AP⊥BCandBQ⊥AC which intersect each other at O. Prove that AO×OP=BO×OQ.

In $Δ A B C, A P ⊥ B C a n d B Q ⊥ A C$ which intersect each other at $O$ . Prove that $A O \times O P = B O \times O Q$ .