Given the rectangle below with length 8 and width 6
a) construct the midpoints of all four sides of the rectangle.
b) connect the midpoints to form another quadrilateral. What type of quadrilateral is formed? Justify your answer.
c) find the area of the quadrilateral you formed.
( There is a picture of a rectangle below. I'm willing to give all my points away.)

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babundra210 babundra210 · Answer 1 · 2020-01-09T12:11:22+01:00

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.