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Top Trapezium (Posted on 2024-04-01)

The circumferences of circles C₁ and C₂ touch at exactly one point.

Both circles also are tangent to a straight line L, at points Q₁ and Q₂ respectively.

The center of C₁ is P₁, and the center of C₂ is P₂.

The radius of C₁ is R and the radius of C₂ is r.

(i) Find the area of the trapezium T with vertices Q₁P₁P₂Q₂ in terms of R and r

(ii) Given that R + r = 10 units, find the maximum area of T.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

| Comment 1 of 3

The trapezoid (trapezium) has parallel bases R and r.

The height is the perpendicular distance which can be find by Pythagorean theorem Q1Q2^2 = (R+r)^2 - (R-r)^2

Q1Q2=2sqrt(Rr)

The area formula is just (R+r)sqrt(Rr)

Intuitively the maximum area would require R=r, forming a rectangle, but using a bit of calculus to confirm:

R=10-r

A(r)=10sqrt(10r-r^2)

A'(r)=-5(10-2r)/sqrt(10r-r^2)

10-2r=0

r=5

A(5)=50

Posted by Jer on 2024-04-02 09:15:02

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