Math Problem Statement

Calculate the exact distance between the points (-2, 3) and (4, 9).

A.)

12 B.)

2 square root of 10 C.)

5 square root of 2 D.)

6 square root of 2

Solution

To calculate the exact distance between the points (2,3)(-2, 3) and (4,9)(4, 9), we use the distance formula:

d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Plugging in the coordinates (2,3)(-2, 3) and (4,9)(4, 9):

x1=2,y1=3x_1 = -2, \quad y_1 = 3 x2=4,y2=9x_2 = 4, \quad y_2 = 9

Calculate the differences:

x2x1=4(2)=4+2=6x_2 - x_1 = 4 - (-2) = 4 + 2 = 6 y2y1=93=6y_2 - y_1 = 9 - 3 = 6

Substitute these into the distance formula:

d=(6)2+(6)2d = \sqrt{(6)^2 + (6)^2} d=36+36d = \sqrt{36 + 36} d=72d = \sqrt{72} d=36×2d = \sqrt{36 \times 2} d=36×2d = \sqrt{36} \times \sqrt{2} d=62d = 6 \sqrt{2}

So, the exact distance between the points is 626 \sqrt{2}.

Therefore, the correct answer is:

D.) 626 \sqrt{2}

Would you like more details or have any questions? Here are some additional questions to expand your understanding:

  1. What is the distance formula used to calculate the distance between two points in a plane?
  2. How do you derive the distance formula from the Pythagorean theorem?
  3. Can the distance formula be applied in three-dimensional space? If so, how?
  4. What are the coordinates of the midpoint between (2,3)(-2, 3) and (4,9)(4, 9)?
  5. How would you calculate the slope of the line segment connecting (2,3)(-2, 3) and (4,9)(4, 9)?
  6. What is the equation of the line passing through the points (2,3)(-2, 3) and (4,9)(4, 9)?
  7. How do you determine if two points are collinear with a third point?
  8. How does the distance formula change if the points are represented in polar coordinates?

Tip: Always simplify your answers to their most reduced form for clarity and precision.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Distance Formula
Square Roots

Formulas

Distance formula

Theorems

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Suitable Grade Level

Grades 9-12