Quadrilaterals · Lesson 8 of 13
The Diagonals of a Rhombus
“Use perpendicular bisection and angle bisection to reason about and construct rhombuses.”
• Explain all three rhombus diagonal properties. • Distinguish angle bisection from segment bisection. • Use half-diagonals and half-angles in calculations. • Construct a rhombus from its diagonals.
When you draw both diagonals inside a rhombus, several equalities appear at once. Each diagonal has two equal pieces, some corner angles are split into equal parts, and the crossing has right angles. These statements can sound similar even though they refer to different things. Looking at them one by one will help you choose the right fact when a question gives a length, a vertex angle or a crossing angle.
You already know that a rhombus is a parallelogram. We can use that result for the midpoint property, then add the extra consequences of having four equal sides.
Because a rhombus is a parallelogram, its diagonals bisect each other. For diagonals AC and BD meeting at O, this means AO = OC and BO = OD. It does not force AC = BD.
Next compare triangles ABC and ADC. AB = AD, BC = DC, and AC is common, so they are congruent by SSS. Thus AC bisects the angles at A and C. Repeating the argument across BD shows that BD bisects the other two vertex angles.
A diagonal bisecting a segment divides its length into equal parts. A diagonal bisecting an angle divides its angle measure into equal parts. Write the relevant equality instead of using the word “bisect” on its own.
Deduction 10 — What can we say about the angles formed by the diagonals of a rhombus at their point of intersection?
Use rhombus GAME, whose diagonals GM and EA meet at O. Compare triangles GEO and MEO. GE = ME because the rhombus sides are equal. GO = MO because its diagonals bisect each other. EO is common.
SSS congruence gives ∠GOE = ∠MOE. Since G, O and M lie on one straight line, those two angles add to 180°. They are equal, so each is 90°. Vertically opposite angles give the same result for the other two angles at O.
The perpendicularity is proved using both side equality and midpoint information. A quadrilateral with merely perpendicular diagonals need not be a rhombus: mutual bisection is also needed for the diagonal construction.
Problem
A rhombus has diagonals 18 cm and 12 cm. Find their four parts.
Problem
A rhombus diagonal makes a 30° angle with one side at V. Find all vertex angles.
Problem
A diagonal bisects a rhombus corner and one part is 20°. Find the adjacent whole angle.
Figure it Out
Problem
Construct a rhombus whose diagonals are 4 cm and 5 cm.
Quiz
The diagonals of a rhombus:
A 14 cm diagonal of a rhombus is split by the other diagonal into:
A rhombus diagonal passes through a 100° vertex angle. It divides it into:
Are rhombus diagonals always equal?
Which conditions can construct a rhombus from two diagonals?
Practice Problems
- A rhombus has diagonals 18 cm and 12 cm. Find each half.
- A diagonal bisects a 70° vertex angle. Find each part.
- A diagonal makes 25° with a side at a rhombus vertex. Find the adjacent whole angle.
- Construct a rhombus whose diagonals are 4 cm and 5 cm.
- What extra diagonal condition makes a rhombus a square?
1. The diagonals bisect each other. 2. Their halves are 9 cm, 9 cm, 6 cm and 6 cm.
1. Angle bisection gives two equal angles. 2. Each is 70° ÷ 2 = 35°.
1. The whole vertex angle is 2 × 25° = 50°. 2. Its adjacent angle is 180° − 50° = 130°.
1. Draw a 5 cm segment and its perpendicular bisector. 2. On that perpendicular, mark points 2 cm on either side of the midpoint. 3. Join consecutive endpoints; congruent small right triangles guarantee four equal sides.
1. A rhombus already has perpendicular diagonals that bisect each other. 2. If the diagonals are also equal, the complete square diagonal test is satisfied. 3. A non-square rhombus therefore does not have equal diagonals.
Key Takeaways
• Rhombus diagonals bisect each other. • They also bisect the vertex angles they pass through. • Their intersection angles are all 90°. • The diagonals need not have equal lengths. • To construct from diagonal lengths, place their midpoints together and make them perpendicular.