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AB is the diameter of a circle with centre O. From a point X on its circumference, a perpendicular XY is dropped on AB such that it intersects AB at Y. If BY = 5 cm and XY = 7 cm, what is the circumference (in cm) of the circle?
Use the property that the altitude to the hypotenuse of a right triangle created by a diameter divides the diameter into two segments whose product equals the square of the altitude.
AB is the diameter of a circle. XY is a perpendicular from a point X on the circumference to AB at Y. BY = 5 cm and XY = 7 cm.
XY2=AY├ЧBY
Use the property that the altitude to the hypotenuse of a right triangle created by a diameter divides the diameter into two segments whose product equals the square of the altitude.
Students often assume the circle's radius is BY, or they forget that the diameter AB=AY+BY, not just AY.
Use Geometric Mean Theorem
In a circle, a perpendicular from a point on the circumference to the diameter creates a geometric mean relationship: the altitude XY squared equals the product of the two segments of the diameter, AY and BY.
XY2=AY├ЧBY
Calculate length of AY
Substitute the given values XY=7 and BY=5 into the equation to find AY.
72=AY├Ч5тЯ╣49=5├ЧAYтЯ╣AY=9.8
Find the Diameter
The diameter AB is the sum of AY and BY.
AB=AY+BY=9.8+5=14.8
Calculate Circumference
The circumference C of a circle is calculated using the formula C=╧А├Чd.
C=14.8╧А
A is correct because the diameter is 14.8 cm, which leads to a circumference of 14.8╧Аcm.
This theorem is a specific case of the Right Triangle Altitude Theorem, which appears frequently in Coordinate Geometry and trigonometry problems involving circles.
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