Shape and Space 4
3 marks · HardExplain, with reference to the properties of parallelograms and right triangles, how you can use Pythagoras’ theorem to prove that the diagonals of a rhombus are perpendicular.
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The diagonals of a rhombus bisect each other at right angles because each diagonal divides the rhombus into two congruent right-angled triangles. As every side of a rhombus is equal and each diagonal acts as the hypotenuse for one triangle and as one of the shorter sides for others, using Pythagoras' theorem shows that the triangles formed are right-angled, making the diagonals perpendicular.
Must show: Each diagonal bisects the rhombus, Diagonals form right triangles with equal sides, Using Pythagoras theorem to show side relationships, Right angle where diagonals cross
Also accept: Each diagonal splits the rhombus into two congruent right triangles, and using Pythagoras’ theorem for these triangles shows the diagonals meet at right angles.
Common mistakes: Confusing with rectangle/other parallelograms; Not mentioning right triangles; Not linking sides to diagonals
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Marking points
M1 · 1 mark · Identifies that diagonals of a rhombus bisect each other and form two congruent right triangles.
M2 · 1 mark · Correctly applies Pythagoras’ theorem to the right triangles formed by the diagonals, relating the side lengths and showing that they create right angles where they cross.
B1 · 1 mark · Concludes or states that this proves the diagonals are perpendicular.
Worked solution
Let the rhombus have sides of length a, and let the diagonals have lengths p and q.
Diagonals of a rhombus bisect each other at point O.
The diagonals divide the rhombus into four right-angled triangles, each with sides p/2 and q/2, and hypotenuse a.
By Pythagoras’ theorem: (p/2)^2 + (q/2)^2 = a^2.
Since both triangles have their right angle at O where the diagonals intersect, the diagonals cross at right angles.