Given - RS and PT are altitudes of - PQR- Prove that -ii PQ - QS - RQ - QT

Step : Proving 𝑃𝑄 × 𝑄𝑆 = 𝑅𝑄 × 𝑄𝑇 Given : RS and PT are altitudes of Δ PQR Now, in Δ PQT and Δ RQS, ∠ nbsp;PQT=∠ RQS nbsp; nbsp; nbsp; nbsp; [Commo ...

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Standard X

Mathematics

Tests for Similarity of Triangles

Given : RS an...

Question

Given : $R S$ and $P T$ are altitudes of $Δ P Q R$ . Prove that :

ii) $P Q \times Q S = R Q \times Q T$

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Solution

Step : Proving $P Q \times Q S = R Q \times Q T$

Given : $R S$ and $P T$ are altitudes of $Δ P Q R$
Now, in $Δ P Q T$ and $Δ R Q S$ ,
$∠ P Q T = ∠ R Q S$ [Common angle]
And $∠ P T Q = ∠ R Q S$
[Both are $90^{\circ}, ∵ R S$ and $P T$ are altitudes]
So, $Δ P Q T$ ∼ $Δ R Q S$ by $A A$ criterion.
$\Rightarrow \frac{P Q}{R Q} = \frac{Q T}{Q S} = \frac{P T}{R S}$
[Corresponding sides of similar triangles are in proportion]
Consider, $\frac{P Q}{R Q} = \frac{Q T}{Q S}$
$\Rightarrow P Q \times Q S = R Q \times Q T$
Hence proved.

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Similar questions

Q. Given :

R S

and

P T

are altitudes of

Δ P Q R

. Prove that :

i)

Δ P Q T

∼

Δ R Q S

Q. In the given figure,

R S = Q T

and

Q S = R T .

Prove that

P Q = P R

In fig. PQRS is a quadrilateral and T and U are respectively points on PS and RS such that
PQ=RQ,
∠PQT=∠RQU...(i)

∠TQS=∠UQS...(ii)
Prove that: QT=QU.

Q. In given figure, IF

\frac{Q T}{P R} = \frac{Q R}{Q S}

AND

∠ 1 = ∠ 2

. Prove that

△ P Q S \sim △ T Q R

In fig, $\frac{Q T}{P R} = \frac{Q R}{Q S}$ and $∠ 1 = ∠ 2$ . Prove that $△ P Q S \sim △ T Q R$ .

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Given : RS and PT are altitudes of ΔPQR. Prove that : ii) PQ×QS=RQ×QT

Given : $R S$ and $P T$ are altitudes of $Δ P Q R$ . Prove that :

ii) $P Q \times Q S = R Q \times Q T$