Prove: (PC)·(QE) + (PD)·(QF) < 8

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In summary, The statement "(PC)·(QE) + (PD)·(QF) < 8" is an inequality that is asking for a proof. It is asking to show that the left side of the inequality is less than 8. The notation (PC)·(QE) + (PD)·(QF) represents the sum of two terms, where the first term is the product of PC and QE, and the second term is the product of PD and QF. The inequality sign < indicates that the left side of the inequality is less than the number 8. The variables PC, QE, PD, and QF represent different quantities or values that are being multiplied together to form the two terms in
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construct OM $\perp CD$
point M is the midpoint of CD
$PC\times QE \leq \dfrac {PC^2+QE^2}{2}----(1)$
$PD\times QF \leq \dfrac {PD^2+QF^2}{2}----(2)$
BUT $PC^2+PD^2$=$(CM-PM)^2+(DM+PM)^2=2(CM^2+PM^2)$
=$2(CM^2+OM^2)=2OC^2=8$
also $QE^2+QF^2=8$
$\therefore (1)+(2)\leq \dfrac {(8+8)}{2}=8$
and the proof is finished
 
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FAQ: Prove: (PC)·(QE) + (PD)·(QF) < 8

What does "Prove: (PC)·(QE) + (PD)·(QF) < 8" mean?

The statement "(PC)·(QE) + (PD)·(QF) < 8" is an inequality that is asking for a proof. It is asking to show that the left side of the inequality is less than 8.

How do you read and interpret this inequality?

The notation (PC)·(QE) + (PD)·(QF) represents the sum of two terms, where the first term is the product of PC and QE, and the second term is the product of PD and QF. The inequality sign < indicates that the left side of the inequality is less than the number 8.

What are the variables PC, QE, PD, and QF in this inequality?

The variables PC, QE, PD, and QF represent different quantities or values that are being multiplied together to form the two terms in the inequality. These variables could represent any numbers or variables that are relevant to the problem at hand.

How can you prove this inequality?

To prove this inequality, you would need to provide a mathematical argument or series of logical steps that show how the left side of the inequality is always less than the number 8. This could involve simplifying the expression, using known mathematical rules and properties, or providing a counterexample.

Why is this inequality important?

Inequalities, like this one, are important because they allow us to compare different quantities or values. In this case, the inequality is asking us to prove that the left side is always less than 8. This could be useful in a variety of situations, such as optimizing a mathematical model or determining the feasibility of a solution to a problem.

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