Diagonal — Geometry, what it means and where it is taught
A segment joining two non-adjacent vertices.
Taught in Geometry.
Where it is taught
What it rests on
Work through these first if this one does not settle.
- Segment — Two endpoints and everything between them. Written AB.
What uses it later
This idea is needed for:
- Identity matrix — The square matrix with ones down the diagonal; it leaves everything unchanged.
Practise this idea
13 questions in the course bank use Diagonal, in Plane Sections of 3-D Figures, Problem Solving, Pythagorean Theorem, Distance on the Coordinate Plane, Quadrilaterals & Polygon Properties, Coordinate Geometry Proofs. The first 8 are below; the rest are in those units.
Problem 1
A cube is sliced through the diagonal of its top face and the parallel diagonal of its bottom face. What shape is the cross section?
- A square
- A rectangle
- A triangle
- A hexagon
Worked answer
A rectangle
The plane crosses four faces, so the cross section has four sides
All four corners are right angles, but the horizontal sides are diagonals — longer than the vertical edges
So it is a rectangle, not a square.
Problem 2
A cube is sliced through the three vertices next to one corner. What kind of triangle appears?
- A right triangle
- An equilateral triangle
- A scalene triangle
Worked answer
An equilateral triangle
Each of the three sides is a diagonal of a square face
All the faces are identical, so all three diagonals are equal
The cross section is an equilateral triangle.
Problem 3
How many diagonals does a hexagon have?
Worked answer
9
Each vertex connects to 6 - 3 = 3 others by a diagonal
6 × 3 = 18, counting each diagonal twice
18 ÷ 2 = 9
Problem 4
A rectangle measures 6 by 8. Find the length of its diagonal.
Worked answer
10
Legs 6 and 8
36 + 64 = 100, so the diagonal is 10.
Problem 5
A square has side 10. Find its diagonal to the nearest tenth.
Worked answer
14.1
100 + 100 = 200
√200 ≈ 14.14
≈ 14.1
Problem 6
A quadrilateral has vertices (0,0), (4,0), (4,4), (0,4). Find the length of a diagonal in simplest radical form.
Worked answer
4sqrt(2)
From (0,0) to (4,4)
16 + 16 = 32
√32 = 4√2
Problem 7
Congruent diagonals that bisect each other mean the shape is a
- rectangle
- rhombus
- trapezoid
- kite
Worked answer
rectangle
Bisecting diagonals make it a parallelogram.
Congruent diagonals upgrade it to a rectangle.
Problem 8
Perpendicular diagonals that bisect each other mean the shape is a
- rhombus
- rectangle
- trapezoid
- square
Worked answer
rhombus
Bisecting makes it a parallelogram.
Perpendicular makes it a rhombus.