Math Problem Statement

Use the distance formula to determine if the figure is a parallelogram. P (-4, 2), Q (6, 4), R (11, -2), S (2, -3)

Solution

To determine if the quadrilateral PQRSPQRS is a parallelogram, we can use the distance formula to calculate the lengths of its sides and check if the opposite sides are equal. In a parallelogram, opposite sides must be congruent (i.e., equal in length).

The distance formula is:

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

Where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of two points.

Step 1: Calculate the lengths of the sides

We will calculate the lengths of the four sides: PQPQ, QRQR, RSRS, and SPSP.

PQPQ:

Using the coordinates of points P(4,2)P(-4, 2) and Q(6,4)Q(6, 4):

PQ=(6(4))2+(42)2=(6+4)2+(42)2=102+22=100+4=104PQ = \sqrt{(6 - (-4))^2 + (4 - 2)^2} = \sqrt{(6 + 4)^2 + (4 - 2)^2} = \sqrt{10^2 + 2^2} = \sqrt{100 + 4} = \sqrt{104}

QRQR:

Using the coordinates of points Q(6,4)Q(6, 4) and R(11,2)R(11, -2):

QR=(116)2+(24)2=(5)2+(6)2=25+36=61QR = \sqrt{(11 - 6)^2 + (-2 - 4)^2} = \sqrt{(5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61}

RSRS:

Using the coordinates of points R(11,2)R(11, -2) and S(2,3)S(2, -3):

RS=(211)2+(3(2))2=(9)2+(1)2=81+1=82RS = \sqrt{(2 - 11)^2 + (-3 - (-2))^2} = \sqrt{(-9)^2 + (-1)^2} = \sqrt{81 + 1} = \sqrt{82}

SPSP:

Using the coordinates of points S(2,3)S(2, -3) and P(4,2)P(-4, 2):

SP=(2(4))2+(32)2=(2+4)2+(32)2=62+(5)2=36+25=61SP = \sqrt{(2 - (-4))^2 + (-3 - 2)^2} = \sqrt{(2 + 4)^2 + (-3 - 2)^2} = \sqrt{6^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61}

Step 2: Compare the lengths of opposite sides

  • PQ=104PQ = \sqrt{104} and RS=82RS = \sqrt{82} (These are not equal).
  • QR=61QR = \sqrt{61} and SP=61SP = \sqrt{61} (These are equal).

Since the opposite sides PQPQ and RSRS are not equal, this is not a parallelogram.

Summary:

The quadrilateral PQRSPQRS does not satisfy the condition of having opposite sides equal, so it is not a parallelogram.

Would you like further details on this or have additional questions? Here are some related questions you might find interesting:

  1. What other methods can be used to determine if a quadrilateral is a parallelogram?
  2. How can we use the slope formula to check if opposite sides are parallel in a quadrilateral?
  3. What is the condition for a quadrilateral to be a rectangle?
  4. How can we prove that two sides are equal using coordinate geometry?
  5. Can we use the distance formula for other types of polygons?

Tip: The distance formula is very useful for verifying side lengths and confirming whether the opposite sides of a quadrilateral are equal when checking for parallelograms or other quadrilaterals!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Distance Formula
Parallelogram Properties

Formulas

Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Parallelogram Properties: Opposite sides are congruent

Suitable Grade Level

Grades 9-11