The distance formula is used to calculate the straight-line distance between two points in the coordinate plane. It is derived from the Pythagorean theorem.

The correct distance formula is derived using the Pythagorean theorem, which leads to d = √[(x₂ − x₝)² + (y₂ − y₝)²]. This expression accurately computes the straight-line distance between two points.

Using the distance formula, the differences in coordinates are 3 and 4. Squaring and adding these values gives 9 + 16 = 25, and the square root of 25 is 5.

In the distance formula, the difference in x-coordinates is squared to ensure that the result is always non-negative. This step is essential for applying the Pythagorean theorem.

The distance formula is directly derived from the Pythagorean theorem, which relates the sides of a right triangle to its hypotenuse. This fundamental theorem underpins many concepts in coordinate geometry.

Subtracting the coordinates gives differences of 6 and 4. Squaring these yields 36 and 16, respectively, and their sum is 52. The square root of 52 simplifies to 2√13.

The difference in x-coordinates is 3 and the difference in y-coordinates is 4. Applying the distance formula results in √(3² + 4²) = √(9 + 16) = √25, which equals 5.

Using the distance formula, the equation simplifies to (6 − x)² + 16 = 25. Solving (6 − x)² = 9 yields 6 − x = 3 or −3, which gives x = 3 or x = 9.

The distance between (1, 2) and (4, 6) is 5 and between (4, 6) and (7, 10) is also 5, while the distance between (1, 2) and (7, 10) is 10. Since the sum of the two smaller distances equals the largest distance, the points are collinear.

The difference in x-coordinates is 6 and in y-coordinates is 8. Squaring these values gives 36 and 64, and their sum is 100. The square root of 100 is 10.

Since both points have the same x-coordinate, the distance is determined solely by the difference in y-coordinates. The calculation |5 - (-5)| results in 10.

The points (0, 0) and (5, 12) have differences of 5 and 12 in their coordinates. The distance calculated is √(5² + 12²) = √(25 + 144) = √169, which equals 13, a well-known Pythagorean triple.

The differences in coordinates are 6 in the x-direction and -8 in the y-direction. Squaring these gives 36 and 64 respectively, and their sum is 100. The square root of 100 is 10.

The distance formula is a direct application of the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse. This provides the basis for measuring the straight-line distance between two points.

Substituting p = 4 into the distance formula gives √[(q - 3)² + (-1 - 4)²] = √[(q - 3)² + 25] = 5. Squaring both sides results in (q - 3)² = 0, so q must equal 3.

By setting the two distance expressions equal and squaring both sides, the resulting equation simplifies to -8k = -56, which yields k = 7. This is the unique value that makes the two distances equal.

The midpoint of A and B is calculated as (4, 5). The distance from this midpoint to A yields √[(4-1)² + (5-2)²] = √(9+9) = √18, which simplifies to 3√2. This is the circle's radius.

The center of the circle is (3, -4) and its radius is 7. The distance from the center to the point (-4, -8) is √[(-7)² + (-4)²] = √(49+16) = √65, which is approximately 8.06 - greater than 7 - so the point lies outside the circle.

Using the distance formula, we set up the equation √[(4 - (-2))² + (3 - k)²] = 10, which simplifies to (3 - k)² = 64. This results in two solutions: k = -5 and k = 11, both of which satisfy the original equation.

The differences in the x- and y-coordinates are 9 and 12, respectively. Squaring these and adding them gives 81 + 144 = 225, confirming that 15² = 225. This demonstrates the validity of the 9-12-15 Pythagorean triple.