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Question Number 4317 by shiv009 last updated on 10/Jan/16

Commented by prakash jain last updated on 10/Jan/16

Vertices of triangle (1,3), (4,−1)   area 5. Find position of 3^(rd) vertex.  (0, 2tan^(−1) (5/4))  (0, tan^(−1) (5/4))  (2tan^(−1) (5/4),0)  none

$$\mathrm{Vertices}\:\mathrm{of}\:\mathrm{triangle}\:\left(\mathrm{1},\mathrm{3}\right),\:\left(\mathrm{4},−\mathrm{1}\right)\: \\ $$$$\mathrm{area}\:\mathrm{5}.\:\mathrm{Find}\:\mathrm{position}\:\mathrm{of}\:\mathrm{3}^{\mathrm{rd}} \mathrm{vertex}. \\ $$$$\left(\mathrm{0},\:\mathrm{2tan}^{−\mathrm{1}} \frac{\mathrm{5}}{\mathrm{4}}\right) \\ $$$$\left(\mathrm{0},\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{5}}{\mathrm{4}}\right) \\ $$$$\left(\mathrm{2tan}^{−\mathrm{1}} \frac{\mathrm{5}}{\mathrm{4}},\mathrm{0}\right) \\ $$$$\mathrm{none} \\ $$

Commented by prakash jain last updated on 10/Jan/16

Please confirm your question.

$$\mathrm{Please}\:\mathrm{confirm}\:\mathrm{your}\:\mathrm{question}. \\ $$

Commented by shiv009 last updated on 10/Jan/16

in this question we have to find that angle of the third  vertex lies in which of the following option.

$${in}\:{this}\:{question}\:{we}\:{have}\:{to}\:{find}\:{that}\:{angle}\:{of}\:{the}\:{third} \\ $$$${vertex}\:{lies}\:{in}\:{which}\:{of}\:{the}\:{following}\:{option}. \\ $$

Commented by prakash jain last updated on 11/Jan/16

A=(4,−1), B(−1,−3)  Length of the base=(√((4−1)^2 +(−1−3)^2 ))  =(√(3^2 +4^2 ))=5  area=(1/2)bh=5⇒h=2  Vertex C will lie between the ⊥^r  to AB drawn on  A and ⊥^r  drawn on C.  Slope of line AB=((−1−(−3))/(4−(−1)))=(2/5)  slope of ⊥^r  line=−(5/2)  You will 4 possible coordinate for C and  vertex has to lie within that region.

$$\mathrm{A}=\left(\mathrm{4},−\mathrm{1}\right),\:\mathrm{B}\left(−\mathrm{1},−\mathrm{3}\right) \\ $$$$\mathrm{Length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{base}=\sqrt{\left(\mathrm{4}−\mathrm{1}\right)^{\mathrm{2}} +\left(−\mathrm{1}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$=\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} }=\mathrm{5} \\ $$$$\mathrm{area}=\frac{\mathrm{1}}{\mathrm{2}}{bh}=\mathrm{5}\Rightarrow{h}=\mathrm{2} \\ $$$$\mathrm{Vertex}\:\mathrm{C}\:\mathrm{will}\:\mathrm{lie}\:\mathrm{between}\:\mathrm{the}\:\bot^{{r}} \:\mathrm{to}\:\mathrm{AB}\:\mathrm{drawn}\:\mathrm{on} \\ $$$$\mathrm{A}\:\mathrm{and}\:\bot^{{r}} \:\mathrm{drawn}\:\mathrm{on}\:\mathrm{C}. \\ $$$$\mathrm{Slope}\:\mathrm{of}\:\mathrm{line}\:\mathrm{AB}=\frac{−\mathrm{1}−\left(−\mathrm{3}\right)}{\mathrm{4}−\left(−\mathrm{1}\right)}=\frac{\mathrm{2}}{\mathrm{5}} \\ $$$$\mathrm{slope}\:\mathrm{of}\:\bot^{{r}} \:\mathrm{line}=−\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$\mathrm{You}\:\mathrm{will}\:\mathrm{4}\:\mathrm{possible}\:\mathrm{coordinate}\:\mathrm{for}\:\mathrm{C}\:\mathrm{and} \\ $$$$\mathrm{vertex}\:\mathrm{has}\:\mathrm{to}\:\mathrm{lie}\:\mathrm{within}\:\mathrm{that}\:\mathrm{region}. \\ $$

Commented by shiv009 last updated on 10/Jan/16

i got it upto this but which option is correct  i did not get .please help me in it.

$${i}\:{got}\:{it}\:{upto}\:{this}\:{but}\:{which}\:{option}\:{is}\:{correct} \\ $$$${i}\:{did}\:{not}\:{get}\:.{please}\:{help}\:{me}\:{in}\:{it}. \\ $$

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