Question

PQRSTU is an artificial ski-slope where PQRS and RSTU are both rectangles and perpendicular to each other. Find: the angle between MP and plane PQRS if M is the midpoint of TU.

Answer 1

To find the angle between the line MP and the plane PQRS, we identify the angle that MP makes with a line that is perpendicular to PQRS. Since MP is perpendicular to RSTU (parallel to the normal), the desired angle is the same as MP's angle with the normal to PQRS.

Step-by-step explanation:

The question pertains to finding the angle between a line and a plane, specifically the line MP and the plane PQRS given that PQRS and RSTU are rectangles perpendicular to each other and M is the midpoint of TU. To find the angle between a line and a plane, we need to consider the angle that the line makes with a line perpendicular to the plane, which is complementary to the desired angle.

As the rectangles PQRS and RSTU are perpendicular to each other, and M is the midpoint of TU, line MP will be perpendicular to plane RSTU. Therefore, the angle we are seeking will be the same as the angle MP makes with the normal to plane PQRS (since the normal to PQRS would be parallel to TU and hence MP in this case).

Using a protractor in practical scenarios, one might measure the angle between a line and a plane by constructing a perpendicular line to the plane from the line in question and measuring the angle between those two lines, where the angle required would be 90 degrees minus the angle measured, accounting for the complementary nature of these angles.

Answer 2

The angle between MP and plane PQRS is 90 degrees.

To find the angle between MP and plane PQRS, we need to understand the geometry of the given ski-slope.

Since PQRS and RSTU are rectangles and perpendicular to each other, we can visualize the ski-slope as a three-dimensional shape.

First, let's consider the rectangle PQRS. The sides PQ and RS are parallel to each other, while the sides PS and QR are perpendicular to PQ and RS.

Next, let's consider the rectangle RSTU. The sides RS and TU are parallel to each other, while the sides RT and SU are perpendicular to RS and TU.

Now, we are given that M is the midpoint of TU. This means that the line segment MP passes through the midpoint M of TU and is perpendicular to it.

Since MP is perpendicular to TU, it is also perpendicular to the side RS of rectangle RSTU.

Since RS is perpendicular to both PQ and MP, it means that MP is also perpendicular to plane PQRS.

Therefore, the angle between MP and plane PQRS is 90 degrees.

In summary, the angle between MP and plane PQRS is 90 degrees because MP is perpendicular to the side RS of rectangle RSTU, which is also perpendicular to plane PQRS.