Answer:
OP = x = 5
Step-by-step explanation:
In the given circle with center O, the radius OQ serves as the perpendicular bisector of chord AB, intersecting at point D. If we connect the center O with point B, we create a right triangle ODB where m∠ODB = 90°.
Given that AB = 8 cm, then DB = 4 cm.
Given that the radius = x cm and DQ = 2 cm, then OD = (x - 2) cm.
To find the value of x, we can use the Pythagorean Theorem:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Pythagorean Theorem}}\\\\a^2+b^2=c^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}[/tex]
In this case:
- a = DB = 4
- b = OD = x - 2
- c = x
Substitute the values into the formula:
[tex]4^2+(x-2)^2=x^2[/tex]
Solve for x:
[tex]16+x^2-4x+4=x^2\\\\x^2-4x+20=x^2\\\\-4x+20=0\\\\4x=20\\\\x=5[/tex]
Therefore, the value of x is x = 5.
