Question 8 of 70intermediate🔍 AnalyzeLong Answer4 marks

Without using distance formula, show that points (−2,−1), (4,0), (3,3)(-2, -1),\ (4, 0),\ (3, 3) and (−3,2)(-3, 2) are the vertices of a parallelogram.

Correct Answer

Let the points be A(−2,−1)A(-2, -1), B(4,0)B(4, 0), C(3,3)C(3, 3) and D(−3,2)D(-3, 2). We will show that opposite sides are parallel using the slope formula.

Slope of AB:

mAB=0−(−1)4−(−2)=16m_{AB} = \frac{0 - (-1)}{4 - (-2)} = \frac{1}{6}

Slope of BC:

mBC=3−03−4=3−1=−3m_{BC} = \frac{3 - 0}{3 - 4} = \frac{3}{-1} = -3

Slope of CD:

mCD=2−3−3−3=−1−6=16m_{CD} = \frac{2 - 3}{-3 - 3} = \frac{-1}{-6} = \frac{1}{6}

Slope of DA:

mDA=−1−2−2−(−3)=−31=−3m_{DA} = \frac{-1 - 2}{-2 - (-3)} = \frac{-3}{1} = -3

Since slope of AB = slope of CD = 16\frac{1}{6}, AB is parallel to CD. Similarly, slope of BC = slope of DA = −3-3, so BC is parallel to DA. Thus, both pairs of opposite sides are parallel, which is a defining property of a parallelogram.

∴ The given points are the vertices of a parallelogram.

Exercise: EXERCISE 9.1 | Q: 8 | (Page 8)
For More Understanding

Explanation

The solution uses the property that a quadrilateral is a parallelogram if its opposite sides are parallel. The slope formula, which is part of coordinate geometry and provided in the context, is applied to each side.

By showing that slopes of AB and CD are equal, and slopes of BC and DA are equal, we establish parallelism. This approach avoids the distance formula entirely, as required by the question.

Solution Steps

  1. Step 1: Label the points as A(-2,-1), B(4,0), C(3,3), D(-3,2).

  2. Step 2: Compute slope of AB = (0 - (-1))/(4 - (-2)) = 1/6.

  3. Step 3: Compute slope of BC = (3 - 0)/(3 - 4) = -3.

  4. Step 4: Compute slope of CD = (2 - 3)/(-3 - 3) = 1/6.

  5. Step 5: Compute slope of DA = (-1 - 2)/(-2 - (-3)) = -3.

  6. Step 6: Conclude AB ∥ CD and BC ∥ DA, so ABCD is a parallelogram.

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