Find the area of the figure by joining the mid points of the adjacent sides of a Rhombus with diagonals 6cm and 8cm.
Explain.
Answers
Answer:
Required area of the formed figure is 12 cm^2.
Step-by-step explanation:
Let a rhombus PQSR, in which length of the diagonals is 6 cm and 8 cm.
We know that the diagonals of rhombus bisect each other at 90°, by Pythagoras theorem.
= > ( half of 1st diagonal )^2 + ( half of 2nd diagonal )^2 = side^2
= > ( 6 cm / 2 )^2 + ( 8 cm / 2 )^2 = side^2
= > ( 3 cm )^2 + ( 4 cm )^2 = side^2
= > 9 cm^2 + 16 cm^2 = side^2
= > 25 cm^2 = side^2
= > ( 5 cm )^2 = side^2 [ Side can't be negative ]
= > 5 cm = side
Now, length of sides of rhombus is 5 cm.
Therefore, PQ = QS = PR = RS = 5 cm.
Thus,
length of PB = length of PA= 5 cm / 2 = 2.5 cm.
Let PS is the diagonal which is 6 cm or 8 cm in length. By mid point theorem, length of AC should be 3 cm or 4 cm. By Pythagoras theorem,
Altitude of ∆AQC = √{ ( 2.5 cm )^2 - ( 2 cm )^2 } or √{ ( 2.5 cm )^2 - ( 1.5 cm )^2 = > 1.5 cm or 2 cm.
Hence, area of AQC is 1 / 2 x 4 x 1.5 cm^2 or 1 / 2 x 3 x 2 = > 3 cm^2 or 3 cm^2 cm^2. Similarly, area of two triangles should be 3 cm^2 and area of other two triangles should be 3 cm^2, including this one.
Thus,
= > Area of the formed figure = ( 1 / 2 x 6 x 8 cm^2 ) - 2( 3 cm^2 + 3 cm^2 ) = 24 cm^2 - 12 cm^2 = 12 cm^2
Hence the required area of the formed figure is 12 cm^2.
Answer:
12 cm
Step-by-step explanation:
Join the mid-points of AB,BC, CD,DA and name them P,Q,R, and S to form a figure PQRS.
ABCD is a rhombus with AC = 8 cm and BD = 6 cm.
Note:
If a triangle and parallelogram are on the same base and between the same parallels, the area of Δ is equal to one-half of the parallelogram.
(i) In ΔABD:
S is mid-point of AD and P is mid-point of AB.
∴ SP ║ BD
⇒ SP = (1/2) BD
⇒ SP = 3 cm.
(ii) In ΔBCD:
R is the mid-point of DC and Q is mid-point of BC.
∴ RQ ║ BD
⇒ RQ = (1/2) BD
⇒ RQ = 3 cm.
∴ SP ║ BD and RQ ║ BD
⇒ SP ║ RQ
Also, SP = RQ = 3 cm.
∴ SPQR is a parallelogram.
Similarly, PQ = SR = (1/2)AC = 4 cm
Now,
BD ║ SP ⇒ SN ║ OM
AC ║ RS ⇒ ON ║ MS
⇒ NOMS is a parallelogram.
∠AOD = 90° {Diagonals of a parallelogram bisect at right angle.
∠MSN = 90° {∵NOMS is a parallelogram}
⇒ SPQR is a rectangle.
∴ Area of SPQR = SP * QP
= 3 * 4
= 12 cm.
Hope it helps!
