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  • In triangle ABC is right triangle with angle B=90 degree . P and Q are midpoints of AB and BC. then prove that: (i) 4 PC²=4 BC² + AB²(ii) 4 AQ² = 4 AB² + BC²(iii) 4 PC² + 4 AQ² = 5 AC²

In triangle ABC is right triangle with angle B=90 degree . P and Q are midpoints of AB and BC. then prove that: (i) 4 PC²=4 BC² + AB²(ii) 4 AQ² = 4 AB² + BC²(iii) 4 PC² + 4 AQ² = 5 AC²

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Given;that;AB=2PB=2AP;and;BC=2BQ=2QC\*(1);4PC^2=4BC^2+AB^2\*In; riangle PBC\*LHS=4PC^2=4(PB^2+BC^2) \*Rightarrow 4(fracAB^24+BC^2)\*Rightarrow 4BC^2+AB^2\* Hence,;proved;4PC^2=4BC^2+AB^2\* (2);4AQ^2=4AB^2+BC^2\*In;triangleABQ\*LHS=4AQ^2=4(AB^2+BQ^2)\*Rightarrow 4(AB^2+fracBC^24)\* Rightarrow 4AB^2+BC^2\* Hence,;proved;4AQ^2=4AB^2+BC^2\* (3);4PC^2+4AQ^2=5AC^2\*LHS=4PC^2+4AQ^2\*Rightarrow 4(BC^2+PB^2)+4(AB^2+BQ^2)\* Rightarrow 4(BC^2+fracAB^24)+ 4(AB^2+fracBC^24)\* Rightarrow 4BC^2+AB^2+BC^2+4AB^2\* Rightarrow 5(AB^2+BC^2)\* herefore In; riangle ABC\* AC^2=AB^2+BC^2\* Rightarrow 5(AC^2)\* Hence,;proved;4PC^2+4AQ^2=5AC^2

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Deependra Verma

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