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Examoogle • User • info@examoogle.com • EE-2024-8821
Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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PQ is a direct common tangent of two circles with centers, O1 and O2, respectively, that contact each other externally at a point A. The circles have radii of 11 cm and 4 cm, respectively. Find the length of PQ.
211 cm
311 cm
411 cm
511 cm
411 cm
For circles touching externally, the direct common tangent length is twice the geometric mean of the radii, simply 2 times the square root of (11 multiplied by 4).
Two externally touching circles have radii r1 = 11 cm and r2 = 4 cm.
L=2r1×r2
For circles touching externally, the direct common tangent length is twice the geometric mean of the radii, simply 2 times the square root of (11 multiplied by 4).
Confusing the formula for direct common tangent (2r1r2) with the formula for transverse common tangent (2d2−(r1+r2)2).
Identify given parameters
The radii of the two circles are r1=11 cm and r2=4 cm. Since they touch externally, the distance between centers d=r1+r2=15 cm.
r1=11,r2=4
Apply Direct Common Tangent Formula
The length L of a direct common tangent for two circles touching externally is given by L=2r1×r2.
L=2r1×r2
Calculate the Result
Substitute the radii values into the formula: L=211×4=244. Simplifying the square root, 44=4×11=211. Thus, L=2×211=411 cm.
L=411
C is correct because the direct common tangent length calculated using 2r1r2 is 411 cm.
This concept is frequently tested alongside the distance between centers of intersecting circles or internal common tangents in competitive geometry.