Math Problem Statement

Amar drew the coordinates of the endpoints of QR¯¯¯¯¯¯¯¯QR¯ at Q (8, 2)Q (8, 2) and R (5, 7)R (5, 7).

Which measurement is closest to the length of QR¯¯¯¯¯¯¯¯QR¯ in units Amar drew on the coordinate plane?

A.5.8 units5.8 units

B.5 units5 units

C.4 units4 units

D.3.9 units3.9 units

Solution

To find the length of QRQR, we can use the distance formula for two points Q(x1,y1)Q(x_1, y_1) and R(x2,y2)R(x_2, y_2), which is:

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

Given the points Q(8,2)Q(8, 2) and R(5,7)R(5, 7), we substitute into the formula:

d=(58)2+(72)2d = \sqrt{(5 - 8)^2 + (7 - 2)^2}

Simplifying:

d=(3)2+(5)2=9+25=34d = \sqrt{(-3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34}

d5.83unitsd \approx 5.83 \, \text{units}

Therefore, the closest measurement to the length of QRQR is:

A. 5.8 units

Would you like further details or clarifications?

Here are 5 related questions:

  1. How would you find the midpoint of segment QRQR?
  2. What is the slope of the line passing through QQ and RR?
  3. Can you determine the equation of the line containing QRQR?
  4. How would the length of the segment change if QQ and RR were moved closer?
  5. Can the distance formula be used in three dimensions?

Tip: The distance formula is derived from the Pythagorean Theorem, where the difference in xx- and yy-coordinates forms the legs of a right triangle.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Distance Between Two Points

Formulas

Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 6-8