Quadrilaterals · Lesson 6 of 13
Parallelogram Diagonals
“Explore the central bisection property of parallelogram diagonals and use it in construction and reasoning.”
• Prove that parallelogram diagonals bisect each other. • Separate bisection from equality and perpendicularity. • Construct a parallelogram from diagonal data. • Justify the reverse diagonal test.
The diagonals of a rectangle are equal and meet at their midpoints. When the frame becomes a slanted parallelogram, which part of that statement survives? Draw a long slanted example and measure both diagonals: one may be much longer than the other. Now locate their crossing point. The useful pattern is about the two pieces of each individual diagonal, not about making the two whole diagonals equal.
We will prove that pattern using triangles on opposite sides of the crossing. Then you can use it to construct a parallelogram even when only its diagonal lengths and crossing angle are known.
Deduction 8 — What is the point of intersection of the two diagonals in a parallelogram?
Use parallelogram EASY, named E → A → S → Y around the boundary. Its diagonals ES and AY meet at O. Compare triangles AOE and YOS. The sides AE and YS are equal because they are opposite sides.
Since AE ∥ YS, the diagonal ES gives ∠AEO = ∠YSO as alternate angles. The diagonal AY gives ∠EAO = ∠SYO. The included sides AE and YS are equal, so the triangles are congruent by ASA.
Corresponding sides give AO = YO and EO = SO. This proves that O is the midpoint of both diagonals. Keep the correspondence A ↔ Y, O ↔ O and E ↔ S; changing the triangle order can incorrectly pair different half-diagonals.
Bisection means AC is divided into two equal parts and BD is divided into two equal parts. It does not say a half of AC equals a half of BD. For diagonals 10 cm and 6 cm, the halves are 5 cm and 3 cm.
A general parallelogram has no fixed diagonal crossing angle and no requirement that the diagonals be equal. Rectangles satisfy the extra equal-length condition; other special cases will add different conditions.
Problem
Diagonals AC = 18 cm and BD = 12 cm meet at O in a parallelogram. Find all four pieces.
Problem
AO = 4.5 cm and BO = 3 cm in parallelogram ABCD. Find AC and BD.
Why does the construction also work in reverse? Suppose AC and BD bisect each other at O. Triangles AOB and COD have equal paired half-diagonals and equal vertically opposite angles, so they are congruent by SAS. Their matching angles establish AB ∥ CD.
Repeating with triangles AOD and COB gives AD ∥ BC. Both pairs of opposite sides are parallel, so the outer quadrilateral is a parallelogram. This explains the construction instead of relying on its appearance.
Figure it Out
Problem
Construct a parallelogram with diagonals 7 cm and 5 cm meeting at 140°.
Quiz
The diagonals of every parallelogram:
AC = 20 cm in a parallelogram and O is the diagonal intersection. AO equals:
Do all parallelogram diagonals have equal lengths?
A parallelogram has diagonal parts AO = 3 cm and OC = 5 cm. What can you conclude?
If the diagonals of a quadrilateral bisect each other, it must be a:
Practice Problems
- A parallelogram has diagonals 24 cm and 10 cm. Find their four halves.
- AO = 7 cm and BO = 4 cm in parallelogram ABCD. Find AC and BD.
- Construct a parallelogram with diagonals 8 cm and 6 cm crossing at 120°.
- A drawing has AO = 3 cm, OC = 5 cm and BO = OD = 4 cm. Can ABCD be a parallelogram?
- What extra diagonal condition makes a parallelogram a rectangle? Explain.
1. Bisect the first: 24 ÷ 2 = 12 cm for each part. 2. Bisect the second: 10 ÷ 2 = 5 cm for each part.
1. AC = 2AO = 14 cm. 2. BD = 2BO = 8 cm.
1. Draw AC = 8 cm and mark midpoint O. 2. Through O draw a line at 120° to OA; mark B and D on opposite sides with OB = OD = 3 cm. 3. Join consecutive endpoints. Both diagonals are bisected, so the quadrilateral is a parallelogram.
1. A parallelogram requires AO = OC as well as BO = OD. 2. Since 3 ≠ 5, the first required equality fails. It cannot be a parallelogram.
1. The parallelogram already supplies mutual bisection. 2. If its diagonals are also equal, they satisfy the full rectangle diagonal test. 3. Perpendicularity is not needed for a rectangle.
Key Takeaways
• Parallelogram diagonals bisect each other. • Half-diagonal equality is checked within each diagonal separately. • The diagonals need not be equal or perpendicular. • Mutual bisection also works as a test for a parallelogram. • For construction, mark half of each given diagonal on either side of the common midpoint.