SOLUTION: In the figure, P and Q are the centers of two tangent circles and the line PQ intersects the circles at Point A and B. The larger circle touches the side CD of the rectangle ABCD a

Algebra.Com
Question 1198770: In the figure, P and Q are the centers of two tangent circles and the line PQ intersects the circles at Point A and B. The larger circle touches the side CD of the rectangle ABCD at point T. If the area of ABCD is 15, what is the area of the triangle PQT?
Found 3 solutions by lotusjayden, ikleyn, math_tutor2020:
Answer by lotusjayden(18)   (Show Source): You can put this solution on YOUR website!
Here is the picture: .
The picture is not drawn to scale, though.
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
In the figure, P and Q are the centers of two tangent circles and the line PQ intersects the circles
at Point A and B. The larger circle touches the side CD of the rectangle ABCD at point T.
If the area of ABCD is 15, what is the area of the triangle PQT?
~~~~~~~~~~~~~~~~~~~~~ 

It is clearly seen from the plot (even by an unarmed eye) that PQ is half of AB,

or, in other words, AB is twice as long as PQ.   (*)


In addition, the height QT of triangle PQT has the same length as the side BC
of the rectangle ABCD.


From it, it is clear that the area of the triangle PQT is 1/4 of the area 
of rectangle ABCD.


ANSWER.  The area of triangle PQT is    square units.

Solved.

-------------

*)   It is written in support to my statement (*) above.

        PQ = r + R;   AB = r + r + R + R = 2*(r+R),   or   AB = 2*PQ.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I'll reference the diagram @lotusjayden has posted.

AB*BC = 15 since the rectangle ABCD has area 15 square units.

AB = AP+PR+RQ+QB
where R is the point of tangency between circles P and Q.

AP = PR as they are radii of circle P
RQ = QB as they are radii of circle Q
So,
AB = AP+PR+RQ+QB
AB = PR+PR+RQ+RQ
AB = 2(PR+RQ)
AB = 2*PQ
PQ = 0.5*AB

Then,
area of triangle PQT = 0.5*base*height
area of triangle PQT = 0.5*PQ*QT
area of triangle PQT = 0.5*(0.5*AB)*BC
area of triangle PQT = 0.25*AB*BC
area of triangle PQT = 0.25*(area of rectangle ABCD)
area of triangle PQT = 0.25*(15)
area of triangle PQT = 3.75 square units
This converts to the improper fraction 15/4.
RELATED QUESTIONS

Prove: If two circles are tangent internally at point P and chord PA of the larger circle (answered by ikleyn)
Three circles with centers A, B, and C are externally tangent to each other as shown in... (answered by Alan3354)
Centers of two circles with radiuses 3cm and 2cm are 20 cm apart. what is the angle... (answered by ikleyn)
Two circles have the same centre and their radii are 15cm and 17cm. A tangent to the... (answered by ikleyn)
Hello I want to prove this T.T but i dont know how I need to draw this diagram and prove... (answered by ikleyn)
Five (5) externally tangent circles have their radius length equal 1 inch and their... (answered by greenestamps)
a) Find the radii of the circles x^2 + y^2 = 2 and (x-3)^2 + (y-3)^2 = 32 b) Find the... (answered by AnlytcPhil)
the centers of two circles of radii 3 and 5 are 17 units apart. find the lengh of a... (answered by robertb,ikleyn)
three circles are tangent to each other externally. The distance between their centers... (answered by Alan3354)