Lines and planes in space

Question 1out of 10

Which lines in space are called skew?

They are the same line

They do not intersect, are not parallel, and are not coplanar

They are parallel in one plane

They intersect at 90°

Question 2out of 10

How many planes pass through three non-collinear points?

Exactly one plane

Infinitely many planes

Exactly two planes

No plane

Question 3out of 10

What relative positions can a line and a plane have?

It always lies in the plane

It may lie in the plane, intersect it once, or be parallel to it

It always has two intersections

It can only be perpendicular

Question 4out of 10

Find the distance between the parallel planes 8x + 15y − 9 = 0 and 8x + 15y + 25 = 0.

The distance between the parallel planes Ax + By + Cz + D₁ = 0 and Ax + By + Cz + D₂ = 0 is |D₂ − D₁| / √(A² + B² + C²).

Question 5out of 10

An oblique segment from point A to a plane is 25 cm long, and its orthogonal projection onto the plane is 24 cm. Find the distance from point A to the plane.

25 cm

31 cm

7 cm

24 cm

The distance from a point to a plane is the length of the perpendicular. The perpendicular, the oblique segment and its projection form a right triangle, with the oblique segment as the hypotenuse. Apply the Pythagorean theorem: if the oblique length is l and the projection is p, then h = √(l² − p²).

Question 6out of 10

When is a line perpendicular to a plane?

It is parallel to the plane

It is perpendicular to two intersecting lines in the plane

It lies in the plane

It is perpendicular to one arbitrary line only

Question 7out of 10

How is the angle between an oblique line and a plane found?

Treat every oblique as perpendicular

Measure any plane angle

Use the perpendicular projection of the oblique segment onto the plane

Ignore the projection

Question 8out of 10

What is a dihedral angle?

The angle between two half-planes with a common edge

The angle at a triangle vertex

A full rotation only

Any angle between two lines

Question 9out of 10

An oblique segment of length 13 has an orthogonal projection of length 12 on a plane. Find the sine of the angle α between the segment and the plane.

5/13

12/5

12/13

5/12

The oblique segment, its orthogonal projection and the perpendicular to the plane form a right triangle. First find the perpendicular length using the Pythagorean theorem. The sine of the angle between the segment and the plane is the perpendicular length divided by the oblique length.

Question 10out of 10

Which statement is a criterion for a line parallel to a plane?

One common point proves parallelism

A line in the plane is outside it

A line parallel to a line in the plane and not lying in the plane is parallel to the plane

Every line is parallel to every plane

Time left: 00:00:00

Finish the test

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Question 10out of 10

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