Home -> Solved problems -> Find the value of the length x
Find the value of the length \(x\)
![](https://mathematicsart.com/wp-content/uploads/2022/01/sqside1.png)
Solution
Let’s recall that a pair of tangents from a point outside a circle have equal length. \(AB=AC\).
![](https://mathematicsart.com/wp-content/uploads/2021/10/sqside3.png)
Applying the previous fact to our problem and knowing that the pink shape is a semi-circle, we get these results
![](https://mathematicsart.com/wp-content/uploads/2021/10/sqside4.png)
![](https://mathematicsart.com/wp-content/uploads/2021/10/sqside5.png)
![](https://mathematicsart.com/wp-content/uploads/2021/10/sqside6.png)
Now, we apply Pythagoras theorem to the next triangle
![](https://mathematicsart.com/wp-content/uploads/2021/10/sqside7.png)
\[(1+y)^{2} =(1-y)^{2}+1^{2}\]
\[y^{2}+2 y+1 =y^{2}-2 y+1+1 \]
\[4 y =1\]
\[y =\frac{1}{4}\]
\[x =y+1 \]
\[=\frac{1}{4}+1 \]
\[=\frac{5}{4} \]
\[\huge x =\frac{5}{4}\]
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