Point C is situated on the circumference of a circle of diameter AB. Show that triangle ACB is right angled triangle. from Physics Motion in A Plane Class 11 Manipur Board
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Physics Part I

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Point C is situated on the circumference of a circle of diameter AB. Show that triangle ACB is right angled triangle.


Let a be the radius of circle. 

Here, OB with rightwards harpoon with barb upwards on top space equals space P with rightwards harpoon with barb upwards on top space a n d space stack O C with rightwards harpoon with barb upwards on top space equals space Q with rightwards harpoon with barb upwards on top

therefore space OA with rightwards harpoon with barb upwards on top space equals space minus straight P with rightwards harpoon with barb upwards on top 

Also, open vertical bar OB with rightwards harpoon with barb upwards on top close vertical bar space equals space open vertical bar stack O C with rightwards harpoon with barb upwards on top close vertical bar equals open vertical bar stack O A with rightwards harpoon with barb upwards on top close vertical bar space equals space a 

Now, from increment AOC comma space we space have 

space space space space space space space space space space space space space space space space space space space space stack O C with rightwards harpoon with barb upwards on top space plus space stack C A with rightwards harpoon with barb upwards on top space equals space stack O A with rightwards harpoon with barb upwards on top

rightwards double arrow space space space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top space equals space stack O A with rightwards harpoon with barb upwards on top space minus space stack O C with rightwards harpoon with barb upwards on top space equals space minus P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top space

 

In increment BOC comma space

space space space space space space space space space space space space space OC with rightwards harpoon with barb upwards on top space plus space stack C B with rightwards harpoon with barb upwards on top space equals space stack O B with rightwards harpoon with barb upwards on top

rightwards double arrow space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top space equals space stack O B with rightwards harpoon with barb upwards on top space minus space stack O C with rightwards harpoon with barb upwards on top space equals negative P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top

N o w comma space space space space space open parentheses stack C A with rightwards harpoon with barb upwards on top close parentheses. open parentheses stack C B with rightwards harpoon with barb upwards on top close parentheses space equals space open parentheses negative P with rightwards harpoon with barb upwards on top minus Q with rightwards harpoon with barb upwards on top close parentheses. open parentheses P with rightwards harpoon with barb upwards on top minus Q with rightwards harpoon with barb upwards on top close parentheses

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus P with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space plus space P with rightwards harpoon with barb upwards on top. Q with rightwards harpoon with barb upwards on top space plus space Q with rightwards harpoon with barb upwards on top. Q with rightwards harpoon with barb upwards on top

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus P squared space plus space Q squared space equals space minus a squared space plus space a squared space equals space 0 space

rightwards double arrow space space space space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top perpendicular stack C B with rightwards harpoon with barb upwards on top
 

Therefore, triangle ACB is a right angled triangle. 

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