Answer:
a) Given in attachment.
b) Quadrilateral formed is a rhombus. Because length of all the sides are equal and angle between adjacent sides is not 90°.
c) 24 unit².
Step-by-step explanation:
Let rectangle be ABCD with length 8 and width 6
And mid-points be E,F,G,H
Now, join the mid-points .
we know that angle between adjacent sides in a rectangle is 90°.
⇒ ΔFAE forms a right angle triangle with 90° at A.
By, pythagoras theorem , EF² = AF² + AE²
⇒ EF² = 3² + 4² = 25
⇒ EF = 5;
In the same way, FG = GH = HE = 5
⇒ the quadrilateral is either square or rhombus.
Now, from ΔFAE, sin(∠AEF) = [tex]\frac{4}{5}[/tex]
(sinФ = opposite side / hypotenuse)
⇒ ∠AEF = 53°;
In similar manner;
∠DEH = 53° . Now ∠FEH = 180 - 53 - 53 = 74° (sum of angles on a straight line is 180°)
⇒ the quadrilateral is rhombus has angle between adjacent sides is 53° and not 90°.
Now, to find the area of rhombus, join EG and FH. And name the intersection point as J.
consider ΔEJF,
it is a right angled triangle at J, and also quadrilateral AFJE forms rectangle, as diagonals intersect perpendicular in a rhombus.
⇒ FJ = 3 and EJ = 4.
⇒ area of this triangle = 1/2 × base × height
= 1/2 × EJ × JF = 1/2 × 3 ×4 = 6
similarly triangles FJG, GJH, HJE.
⇒ area of rhombus = 6+6+6+6 = 24.
