How many methods for this?
Geometry
Level
4
Y intercept(a) - 5 units , X intercept(b) - 7 units
D is the diameter of bigger circle and d diameter of smaller circle .
Find d + D
- Note - The circles are not concentric.
The answer is 12.00.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Let us first focus on the inscribed circle and the right triangle. Lines AB and BC are tangent lines of the circle at points G and E respectively. Thus angle(FGB) and angle(FEB) are both right angles. Lines GF and FE are both the radii of the circle. Hence, GF=FE. Since GF=FE, angle(FGB)=angle(FEB)=angle(GBE)= 90 degrees Therefore, GFEB is a square. Let each side of the square be of length x.
ON TRIANGLES AGF and ADF, a.) GF=FD (both are radii) b.) angle(AGF)= angle(ADF)= 90 degrees (tangent) c.) AF=AF Hence, by Pythagorean theorem, AG= AD = 5-x
SIMILARLY ON TRIANGLES CDF and CEF, CD= EC= 7-x
BY PYTHAGOREAN THEOREM ON TRIANGLE ABC (with AB=5 and BC=7) AC= sqrt(74) AD+DC= sqrt(74) 5-x+7-x = sqrt(74) 12-2x= sqrt(74) 2x= 12-sqrt(74) = d
Now, let us move on to the larger circle with the right triangle inside. We can move the right triangle inside such that point B together with points A and C touches the circle. With that we now have AC as the diameter of the larger circle. AC= sqrt(74) = D
Therefore, d+D= 12-sqrt(74) + sqrt(74) = 12