Diametrically Opposed

Diametrically Opposed

Two points directly opposite each other on a circle or sphere. More formally, two points are diametrically opposed if they are on opposite ends of a diameter.

Circle with two points A and B at opposite ends of a diameter, illustrating diametrically opposed points.

See also

Antipodal points

Key Formula

B = (2h - x_A,\; 2k - y_A)

Where:

$(h, k)$ = Center of the circle
$(x_A, y_A)$ = Coordinates of the known point A on the circle
$B$ = The diametrically opposed point on the circle

Worked Example

Problem: A circle has center (3, 4) and radius 5. Point A is at (7, 7). Find the point B diametrically opposed to A.

Step 1: Verify that A lies on the circle by computing the distance from A to the center.

d = \sqrt{(7 - 3)^2 + (7 - 4)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Step 2: Since the distance equals the radius (5), point A is on the circle. Now apply the formula. The center (h, k) is the midpoint of A and B, so B = (2h − x_A, 2k − y_A).

B = (2(3) - 7,\; 2(4) - 7) = (6 - 7,\; 8 - 7)

Step 3: Simplify to find the coordinates of B.

B = (-1,\; 1)

Step 4: Verify: check that B is also on the circle.

d = \sqrt{(-1 - 3)^2 + (1 - 4)^2} = \sqrt{16 + 9} = 5 \checkmark

Answer: The diametrically opposed point is B = (−1, 1).

Another Example

This example differs by testing whether two given points are diametrically opposed, rather than finding an unknown point. It also highlights that the midpoint condition alone is not sufficient — both points must actually lie on the circle.

Problem: Point P is at (6, 2) and point Q is at (−2, 8). Determine whether P and Q are diametrically opposed on the circle x² + y² − 4x − 10y + 12 = 0.

Step 1: Rewrite the circle equation in standard form by completing the square.

(x^2 - 4x + 4) + (y^2 - 10y + 25) = -12 + 4 + 25

Step 2: Simplify to find the center and radius.

(x - 2)^2 + (y - 5)^2 = 17 \quad \Rightarrow \quad \text{center } (2, 5),\; r = \sqrt{17}

Step 3: For P and Q to be diametrically opposed, their midpoint must equal the center. Find the midpoint of P(6, 2) and Q(−2, 8).

M = \left(\frac{6 + (-2)}{2},\; \frac{2 + 8}{2}\right) = (2,\; 5)

Step 4: The midpoint (2, 5) matches the center. Now verify both points lie on the circle.

(6-2)^2 + (2-5)^2 = 16 + 9 = 25 \neq 17

Step 5: Point P does not lie on the circle, so P and Q are NOT diametrically opposed on this circle, even though their midpoint is the center.

Answer: P and Q are not diametrically opposed on this circle because P does not lie on the circle, despite the midpoint coinciding with the center.

Frequently Asked Questions

What is the difference between diametrically opposed points and antipodal points?

They mean the same thing. 'Antipodal' is the more formal mathematical term used especially for spheres (like opposite poles of the Earth), while 'diametrically opposed' is a more descriptive phrase. Both refer to two points at opposite ends of a diameter.

How do you find the diametrically opposed point on a circle?

Use the center as the midpoint. If the center is (h, k) and one point is (x_A, y_A), the opposite point is (2h − x_A, 2k − y_A). This works because the center lies exactly halfway between any two diametrically opposed points.

What angle does a diameter subtend from a diametrically opposed point?

Any angle inscribed in a semicircle — that is, an angle whose vertex is on the circle and whose sides pass through two diametrically opposed points — is exactly 90°. This is Thales' theorem, one of the most important results in circle geometry.

Diametrically Opposed Points vs. Antipodal Points

	Diametrically Opposed Points	Antipodal Points
Definition	Two points at opposite ends of a diameter of a circle or sphere	Two points at opposite ends of a diameter of a circle or sphere
Common context	Everyday and geometry usage for circles and spheres	Formal math and geography (e.g., opposite poles on Earth)
Formula (2D)	B = (2h − x_A, 2k − y_A)	Same formula applies
Key property	Center is the midpoint of the two points	Center is the midpoint of the two points
Usage note	More descriptive, often used in plain English	More technical, preferred in topology and spherical geometry

Why It Matters

Diametrically opposed points appear throughout circle geometry — from Thales' theorem (any angle inscribed in a semicircle is 90°) to constructions involving perpendicular bisectors. In coordinate geometry, finding the opposite endpoint of a diameter is a standard exam question that tests your understanding of midpoints and circle equations. The concept also extends to real-world contexts like geography (antipodal points on Earth) and physics (opposite poles of a sphere).

Common Mistakes

Mistake: Adding the center coordinates to the known point instead of using the reflection formula.

Correction: The diametrically opposed point is found by reflecting through the center: B = (2h − x_A, 2k − y_A). A common error is computing (h + x_A, k + y_A), which does not produce the correct reflection.

Mistake: Assuming two points are diametrically opposed just because their midpoint equals the center, without checking that both points lie on the circle.

Correction: The midpoint condition is necessary but not sufficient. You must also verify that the distance from each point to the center equals the radius.

Related Terms

Diameter of a Circle or Sphere — The line segment connecting diametrically opposed points
Antipodal Points — Synonym for diametrically opposed points
Circle — The curve on which diametrically opposed points lie
Sphere — 3D surface where the concept extends naturally
Point — The fundamental object being described
Midpoint — The center is the midpoint of any diameter
Radius — Half the diameter; distance from center to each point