Question

The following set of coordinates represents which figure? (7, 10), (4, 7), (6, 5), (9, 8) Parallelogram Rectangle Rhombus Square

Answer 1

The set of coordinates represents a rectangle.

Answer 2

Rectangle

Explanation:

There are four different types of quadrilateral,

A parallelogram has two pairs of opposite equal parallel sides and two pair of opposite equal angle.

A rectangle has two pairs of opposite equal parallel sides with 4 right angles.

A rhombus has 4 equal sides with two pairs of equal opposite angles,

A square has 4 equal sides with 4 right angles,

Now, Let A≡(7, 10), B≡(4, 7), C≡(6, 5), D≡(9, 8)

By the distance formula,

`AB=√((4-7)^2+(7-10)^2)`

`=\sqrt{(-3)^2+(-3)^2`

`=√(9+9)=√(18)=3√(2)\text{ unit}`

Similarly,

BC = √8 unit,

CD = √18 unit,

DA = √8 unit,

Now, the slope of AB,

`m_1=(7-10)/(4-7)`

`=(-3)/(-3)`

`=1`

Similarly, slope of BC = -1

Slope of CD = - 1

And, slope of DA = 1

Since, Slope of AB × Slope of BC = - 1

⇒ AB ⊥ BC

⇒ ∠B = 90°,

Also, Slope of BC × Slope of CD = - 1

⇒ BC ⊥ CD

⇒ ∠C = 90°,

Now, Slope of CD × slope of DA = -1

⇒ CD ⊥ DA

⇒ ∠D = 90°

And, Slope of AB × slope of DA = -1

⇒ AB ⊥ DA

⇒ ∠A = 90°

Since, AB ⊥ BC and BC ⊥ CD

⇒ AB ║ CD

Similarly, AB ⊥ BC and AB ⊥ DA

⇒ BC ║ DA

Hence, by the above definition,

ABCD is a rectangle