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Find the fourth side of a quadrilateral inscribed in a circle having one of its sides equal to 20 m as its diameter, and the other two sides adjacent to the diameter, and the other two sides adjacent to the diameter are 8 m and 12 m respectively.

 Dec 28, 2014

Best Answer 

 #2
avatar+118694 
+10

I am not sure at the moment if Chris and I have interpreted the question the same.

 (This diagram is drawn to scale)

Here is my interpretation.

$$\\

 

Let length BC = x

 

Now I will use the cosine rule:

 

cosA=b2+c2a22bcsocosθ=82+162x22×8×16andcosθ=3362+122x22×336×12Hence82+162x22×8×16=3362+122x22×336×12320x2128=480x212×421320x2128=480x212×421320x28=480x2321321(320x2)=8(480x2)321×320321x2=38408x2960213840=(8+321)x2

 

x2=9602138408+321x=+9602138408+321

 

(960×213840)(8+3×21)=9.864242223858689

 

so 

The forth side is approx 9.86metres long

Since our answers are the same I guess we interpreted the question the same after all     

 Dec 29, 2014
 #1
avatar+130327 
+10

I'm assuming that you're saying that one side of the quadrilateral is the same as the circle's diameter, and we have two adjacent sides to this that are 8 and 12m, respectively. Then the circle's radius = 10 m

To find the remaining side, we need to first find the central angles intercepting the 8 and 12 m sides.....

We can use the Law of Cosines in both cases....for the 12m side, we have

12^2 = 10^2 + 10^2 - 2(10)(10)cosΘ   ......simplifying, we have

cosΘ= 7/25  = cos-1(7/25) = about 73.739795291688°

For the 8m side, we have

8^2 = 10^2 + 10^2 - 2(10(10)cosΘ   ...simplifying again, we have

cosΘ = 17/25 = cos-1(17/25) = about 47.156356956404°

So, since the diameter would span 180° of arc, the remaing side must span (180 - 73.739795291688 - 47.156356956404)° = 59.103847751908° of arc

So...the remaining side... (s).... is given by

s^2  = 10^2 + 10^2 - 2(10)(10)cos(59.103847751908)

s = √[10^2 + 10^2 - 2(10)(10)cos(59.103847751908)] = about 9.86 m

This makes sense....the greater side intercepts the greatest arc, the next greatest side intercepts the next greatest arc, etc.

 

 Dec 28, 2014
 #2
avatar+118694 
+10
Best Answer

I am not sure at the moment if Chris and I have interpreted the question the same.

 (This diagram is drawn to scale)

Here is my interpretation.

$$\\

 

Let length BC = x

 

Now I will use the cosine rule:

 

cosA=b2+c2a22bcsocosθ=82+162x22×8×16andcosθ=3362+122x22×336×12Hence82+162x22×8×16=3362+122x22×336×12320x2128=480x212×421320x2128=480x212×421320x28=480x2321321(320x2)=8(480x2)321×320321x2=38408x2960213840=(8+321)x2

 

x2=9602138408+321x=+9602138408+321

 

(960×213840)(8+3×21)=9.864242223858689

 

so 

The forth side is approx 9.86metres long

Since our answers are the same I guess we interpreted the question the same after all     

Melody Dec 29, 2014

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