Final answer:
The perimeter of quadrilateral BXOY is calculated using the midpoint theorem and the Pythagorean theorem, giving a total length of 14 cm. Thus, the perimeter of quadrilateral BXOY is 14 cm.
Step-by-step explanation:
To find the perimeter of the quadrilateral BXOY in a circle with chords AB and BC that are perpendicular to each other, we need to use the property that the line segment joining the center of a circle to the midpoint of a chord is perpendicular to the chord. Since X and Y are the midpoints of AB and BC respectively, OX is perpendicular to AB and OY is perpendicular to BC. Furthermore, AB and BC are perpendicular to each other, forming right angles at point B.
Since X is the midpoint of AB, XB = ½ × 8 cm = 4 cm. Similarly, YB = ½ × 6 cm = 3 cm. To find OX and OY, we will use the Pythagorean Theorem in the right-angled triangles BOX and BOY. In triangle BOX, OX=√(OB² - XB²) and in triangle BOY, OY=√(OB² - YB²).
To find the length of OB, we know that since AB and BC are perpendicular, triangle ABC is also a right triangle with hypotenuse OC (as OC is the radius for both chords).
We calculate OC using the Pythagorean Theorem with AB and BC: OC = √(AB² + BC²)
= √(8² + 6²) cm = √(64 + 36) cm
= √100 cm = 10 cm.
Now, we have OB = ½ OC = 5 cm.
Now we can find OX and OY: OX = √(5² - 4²) cm = √(25 - 16) cm
= √9 cm = 3 cm, and OY
= √(5² - 3²) cm
= √(25 - 9) cm
= √16 cm
= 4 cm.
To find the perimeter of BXOY, we sum the lengths of its sides: BX + XO + OY + YB = 4 cm + 3 cm + 4 cm + 3 cm
= 14 cm.
Thus, the perimeter of quadrilateral BXOY is 14 cm.