Answer:
Rectangle
Step-by-step 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,
[tex]AB=\sqrt{(4-7)^2+(7-10)^2}[/tex]
[tex]=\sqrt{(-3)^2+(-3)^2[/tex]
[tex]=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\text{ unit}[/tex]
Similarly,
BC = √8 unit,
CD = √18 unit,
DA = √8 unit,
Now, the slope of AB,
[tex]m_1=\frac{7-10}{4-7}[/tex]
[tex]=\frac{-3}{-3}[/tex]
[tex]=1[/tex]
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
