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Radda [10]
2 years ago
10

Question 9

Mathematics
1 answer:
Alenkasestr [34]2 years ago
8 0

Answer:

PX=4\\PY=4\sqrt{3}

Step-by-step explanation:

Let \angle PYZ=Q

\Rightarrow \angle XYP=90-\angle PYZ\\\angle XYP=90-\theta

In the \Delta PYZ

\tan\theta =\frac{opposite}{adjacent}=\frac{PZ}{PY}......(1)

In the \Delta YPX

\tan(90-\theta)=\frac{opposite}{adjacent}=\frac{PX}{PY}

\cot=\frac{PX}{PY}....(2)

eqn(1)\times eqn(2)

\tan\times\cot\theta=\frac{PZ}{PY}\times\frac{PX}{PY}\\\\1=\frac{PZ\times PX}{PY^2}\ \ (as\ tan\theta\ =\frac{1}{\cot\theta})\\\\PY^2=PZ\times PX\\PY^2=12PX.......(3)

Now in \Delta XYP

use Pythagorean theorem

XY^2=PY^2+PX^2\\8^2=PY^2+PX^2\ \ (as\ XY\ =8)\\PX^2+PY^2=64\\\\PY^2=12PX\ \ (eqn(3)\\\\\Rightarrow PX^2+12PX=64\\\\PX^2+12PX-64=0\\\\PX^2+16-4PX-64=0\\\\(PX+16)(PX-4)=0\\\\PX=4,\ -16

Length can not be negative

Hence PX=4

PY^2=12PX=12\times4=48\\\\PY=\sqrt{48}=4\sqrt{3}

Hence PX=4,\ PY=4\sqrt{3}

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