Lines, planes and solids in space · Grade 12 · 4 units
In Lines, planes and solids in space (Grade 12 · 4 units) you practise 5 skills: Two lines in space · Planes in space · A line perpendicular to a plane · The parts of a solid and its area · The diagonal of a box. Two lines in space can lie in one of three ways. Intersecting lines meet at one point. Parallel lines lie in one plane and never meet. Skew lines are not parallel and never meet, and no plane holds them both. In a box, for example, the edges AB and CC′ are skew.
A plane is fixed by three points not on one line, a line and a point not on it, two intersecting lines or two parallel lines. A line can lie in a plane, meet it at one point or be parallel to it. Two different planes are parallel, or meet in one line, like two neighbouring faces of a box, which meet in their common edge. A line is perpendicular to a plane when it is perpendicular to every line in the plane. To show it, two intersecting lines of the plane are enough. In a box, the edge AA′ is perpendicular to the base, because it is perpendicular to the two edges of the base that leave the corner A. In a right pyramid, the height meets the base at the centre of the circle through its corners. A prism has two bases and side faces, and in a right prism the side faces are rectangles. The lateral area of a right prism is the perimeter of the base times the height, and the total surface area adds the two bases. The diagonal of a box is the hypotenuse of a right triangle whose legs are the diagonal of the base and the height.
Tip: Mark in the drawing the right triangle that holds the length you want, and work in it with Pythagoras’ theorem or with trigonometry.
What you learn
Two lines in space
For example: In the box ABCDA′B′C′D′ , what is the relative position of the lines DD′ and BB′ ?
- Intersecting
- Skew
- Parallel
Answer and solution
The answer: Parallel
The lines lie in one plane and never meet, so they are parallel.
Planes in space
For example: Which of these fixes exactly one plane?
- Three points on one line
- Two points
- A line and a point on it
- Two parallel lines
Answer and solution
The answer: Two parallel lines
The right answer is “two parallel lines”: exactly one plane passes through them. No plane passes through two skew lines, and infinitely many planes pass through each of the others.
A line perpendicular to a plane
For example: In the box ABCDA′B′C′D′ , which of these lines is perpendicular to the plane ABCD ?
- CD
- AD
- B′D
- DD′
Answer and solution
The answer: DD′
The edge DD′ is perpendicular to two intersecting lines of the plane, CD and AD , because the faces of a box are rectangles. So DD′ ⊥ ABCD .
The parts of a solid and its area
For example: How many vertices does the box ABCDA′B′C′D′ have?
Answer and solution
The answer: 8
The corners of the two bases: 4 + 4 = 8 .
The diagonal of a box
For example: In the box ABCDA′B′C′D′ the lengths are AB = 3 , BC = 4 and CC′ = 12 . What is the length of the diagonal AC′ ?
Answer and solution
The answer: 13
The diagonal of the base: AC² = 3² + 4² = 25 . In the right triangle ACC′ the diagonal is the hypotenuse: AC′ = √(25 + 12²) = 13 .
Not on the site yet
Parts of this topic that are not built yet: The lateral area of a solid that is not right.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).