Chapter 12.7, Problem 23PPS

a.

To determine

Find whether line CD is perpendicular to line DA or not. Explain

Answer to Problem 23PPS

The line CD is not perpendicular to line DA.

Explanation of Solution

Given:

It is given in the question that there is a quadrilateral ABCD on sphere P. It has four sides such that $DC\perp CB$ , $AB\perp CB$ & $DC\cong AB$ .

  

Concept Used:

In this, use the concept of observing the figure and use your analytical power to judge it.

Calculation:

Since, If it has to be seen in the figure that the lines are not perpendicular. DC and AB are going straight up to the pole but DB follow an arc across it.

Conclusion:

The lines are not perpendicular.

b.

To determine

Find the comparison of DA to CB.

Answer to Problem 23PPS

DA is shorter than CB.

Explanation of Solution

Given:

It is given in the question that there is a quadrilateral ABCD on sphere P. It has four sides such that $DC\perp CB$ , $AB\perp CB$ & $DC\cong AB$ .

  

Concept Used:

In this, use the concept of observing the figure and use your analytical power to judge it.

Here, if seeing the figure of the spherical geometry, it has seen that DA is shorter than CB. As DB gets closer to the pole its length goes towards zero.

Conclusion:

DA is shorter than CB.

c.

To determine

Find whether a rectangle as defined in Euclidean geometry exist in non − Euclidean geometry. Explain

Answer to Problem 23PPS

A rectangle will not exist in non − Euclidean geometry.

Explanation of Solution

Given:

It is given in the question that there is a quadrilateral ABCD on sphere P. It has four sides such that $DC\perp CB$ , $AB\perp CB$ & $DC\cong AB$ .

  

Since, a rectangle as defined in Euclidean geometry cannot exist in spherical geometry. The smaller the side lengths the closer you can get to four right angles but your rectangle will go to a point before you get four right angles.