Vectors

How do you prove diagonals of a parallelogram bisect each other using vectors?

Quick answer

To prove the diagonals bisect each other, set two adjacent sides as base vectors u and v. Express the intersection point P along diagonal AC as s(u + v) and along diagonal DB as v + t(u - v). Equating both expressions gives s*u + s*v = t*u + (1 - t)v. Comparing coefficients yields s = 1/2 and t = 1/2, proving both diagonals meet at their midpoints.

The topic explained

Set up base vectors for the figure

To prove geometric properties using vectors, you first choose one vertex as your reference origin. For parallelogram ABCD, make vertex A the origin and define two non-parallel edge vectors: vector AB = u and vector AD = v. Because opposite sides of a parallelogram are equal and parallel, vector DC = u and vector BC = v.

Every other segment in the figure can now be written as a linear combination of u and v. By vector addition, diagonal AC = AB + BC = u + v. Diagonal DB runs from D to B, which equals DA + AB = -v + u = u - v.

Express the diagonal intersection point in two ways

Let the two diagonals intersect at point P. Because P lies on diagonal AC, vector AP is a scalar multiple of vector AC. We can write vector AP = s * (u + v), where s is an unknown number between 0 and 1.

Point P also lies on diagonal DB. Starting at vertex A, we can reach P by traveling along AD and then partway along DB: vector AP = AD + DP = v + t * (u - v), where t is an unknown number between 0 and 1. Expanding and grouping like vectors gives vector AP = t * u + (1 - t) * v.

Compare coefficients to solve for the ratios

Because u and v are linearly independent, a vector can be written in terms of u and v in exactly one way. That means we can equate the coefficients of u and the coefficients of v from both expressions for vector AP: s = t for vector u, and s = 1 - t for vector v.

Substitute s for t in the second equation: s = 1 - s, which simplifies to 2s = 1, so s = 1/2. Because s = t, t is also 1/2. Since s = 1/2, point P lies exactly halfway along AC. Since t = 1/2, point P lies exactly halfway along DB. This proves that the diagonals bisect each other.

Free printable worksheet

Vector Proofs in Quadrilaterals Worksheet

Designed for 12th grade math students. Express all segments in terms of the given base vectors, set up linear combinations, and equate coefficients to find unknown ratios.

Print the worksheet Regenerate new worksheet

Free, no sign-up. The answer key prints on the second page. Regenerating a new worksheet on the same topic needs a free account.

Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Vector Proofs in Quadrilaterals WorksheetName:

In parallelogram ABCD, vector AB = u and vector AD = v. Express the diagonal vector AC in terms of u and v.
In parallelogram ABCD, vector AB = u and vector AD = v. Express the diagonal vector BD in terms of u and v.
Point M is the midpoint of side BC in parallelogram ABCD with AB = u and AD = v. Express vector AM in terms of u and v.
In trapezium ABCD, base AB is parallel to base DC, with DC = 2 * AB. Let vector AB = u and vector AD = v. Express diagonal vector AC in terms of u and v.
Using the same trapezium ABCD (AB = u, AD = v, DC = 2u), express vector BC in terms of u and v.
In trapezium ABCD with AB parallel to DC, M is the midpoint of AD and N is the midpoint of BC. Prove vector MN is parallel to the bases by expressing MN in terms of AB and DC.
In parallelogram ABCD, point E lies on BC such that BE:EC = 1:2. Segment AE intersects diagonal BD at point P. Express vector AP in terms of u and v, and find the ratio in which P divides BD.

Common questions

Why must the two base vectors be non-parallel?

If base vectors are parallel, one is a multiple of the other, making them linearly dependent. Equating coefficients only works when vectors point in distinct directions, ensuring each point has a unique combination.

How do you prove the midline of a trapezium using vectors?

Define base vectors along one side and the parallel bases. Write the midline vector connecting the midpoints of the non-parallel legs. Adding the leg vectors shows the midline vector equals half the sum of the two base vectors.

What is the common mistake when comparing coefficients?

The most common mistake is forgetting to group all terms for each base vector before comparing. Make sure each side has the format a*u + b*v before setting corresponding scalar components equal to one another.

Can this method find where lines intersect outside the figure?

Yes. The same linear combination method applies. When lines intersect outside the original segment, the scalar values simply turn out to be negative or greater than 1.

How does Lessi help teach vector proofs?

Lessi is an AI tutor that walks through vector geometry step by step. You can practice setting up vectors, and the first 60 minutes of lessons with Lessi are free with an account.

Still have a question? Ask Lessi

Any question about this subject. Lessi answers right here, step by step.

Try asking

Asking needs a free account; your question is kept while you sign up. Answers are written by AI and can contain mistakes. Lessi doesn’t give medical or legal advice.