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3) In the diagram, two circles touch at P. Also, QP and SU are perpendicular diameters of

the larger circle that intersect at O. Point V is on QP and VP is a diameter of the

smaller circle. The smaller circle intersects SU at T, as shown.

If QV = 9 and ST = 5, what is the sum of the

lengths of the diameters of the two circles?

3 In the diagram two circles touch at P Also QP and SU are perpendicular diameters of the larger circle that intersect at O Point

Answer :

The sum of the lengths of the diameters of the two circles is 28.

To find the sum of the lengths of the diameters of the two circles, let:

D1 = the diameter of the larger circle

D2 = diameter of the smaller circle

Given:

QV = 9

ST = 5.

Since, QP is a diameter, the distance from Q to P (the radius) is equal to the diameter of the larger circle divided by 2.

So, QP = D1 and QV + VP = D2.

Because QP and SU are perpendicular and intersect at O, it means that the radius from O to P (on QP) and O to T (on SU) will give the lengths of the radii of the circles.

The distance ST is part of the diameter of the smaller circle:

If ST = 5, then the distance from S to the center of the smaller circle (at point V) can help in determining D2.

We form equations:

D1 = 2 * QP = 2 * (QV + VP) (where VP = the radius of the smaller circle).

Because QV = 9, the full length of the larger diameter is based on the sum of these distances.

We can determine D1:

Since QV + VP = D2 and QV = 9, we will find VP.

If ST = 5, it means that this is part of the smaller circle, and since VP is perpendicular and intersects, we can conclude it contributes to the total.

The diameter of the larger circle will contribute D1 = 2 * 9 = 18.

For the smaller circle, if ST = 5, it implies the radius leads to D2 = 2 * 5 = 10.

Total sum of the diameters = D1 + D2 = 18 + 10 = 28.

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