Let O, P, and Q be distinct collinear points with PQ=3. Let P′ and Q′ be the images of points P and Q under dilation from point O with scale factor 4. What is the length of the image segment P′Q′?

Given:

O, P, and Q be distinct collinear points with PQ=3.

P′ and Q′ be the images of points P and Q under dilation from point O with scale factor 4.

To find:

The length of the image segment P'Q'.

Solution:

We know that,

P′ and Q′ be the images of points P and Q under dilation from point O with scale factor 4. So, Segment P'Q' is the image of segment PQ and the scale factor is 4.

It is given that PQ=3.

Multiply both sides by 3.

Therefore, the length of the image segment P'Q' is 12 units.

the roman republic began to expand when rome fought the punic wars and defeated carthage. the punic wars refer to the series of 3 wars fought among rome and carthage, before it ended in carthage's defeat.

the answer is b. apsidal precession