In a rectangle ABCD the lengths of sides AB, BC, CD and DA are (5x + 2y + 2)cm, (x + y + 4) cm, (2x + 5y – 7) cm and (3x + 2y – 11) cm respectively. Which of the following statement is/are true?
(A) one of the sides of the rectangle is 15 cm long
(B) each diagonal of the rectangle is 38 cm long
(C) perimeter of the rectangle is 100 cm
(D) area of the rectangle is 560 cm2
Answers
Answer:
[0ption] A is the correct statement.
Step-by-step explanation:
According to the question, ABCD is a rectangle with :
➠ AB = (5x + 2y + 2)cm
➠ BC = (x + y + 4)cm
➠ CD = (2x + 5y – 7)cm
➠ DA = (3x + 2y – 11)cm
We know that opposite sides are equal in a rectangle. So :
➠ AB = CD
➠ BC = DA
Let's equate them :
For AB = CD :
➠ 5x + 2y + 2 = 2x + 5y – 7
➠ 5x – 2x + 2y – 5y = – 7 – 2
➠ 3x – 3y = – 9
➠ x – y = – 3 . . . . . ⓵
For BC = DA :
➠ x + y + 4 = 3x + 2y – 11
➠ x – 3x + y – 2y = – 11 – 4
➠ – 2x – y = – 15 . . . . . ⓶
Getting the value of 'x' from ⓵ :
➠ x – y = – 3
➠ x = – 3 + y . . . . . ⓷
Substituting this value of 'x' from ⓷ in ⓶ :
➠ – 2x – y = – 15
➠ – 2 (– 3 + y) – y = – 15
➠ 6 – 2y – y = – 15
➠ – 3y = – 15 – 6
➠ y = – 21/– 3
➠ y = 7
Substituting the value of 'y' in ⓵ :
➠ x – y = – 3
➠ x – 7 = – 3
➠ x = – 3 + 7
➠ x = 4
Now, on the values of 'x' and 'y' substituting in the sides we get :
➠ AB = CD = 36cm
➠ BC = DA = 15cm
Let's calculate the area of the rectangle :
➠ Area of rectangle = Length × Breadth
➠ 36cm × 15cm
➠ Area = 540cm²
Now, let's calculate the perimeter of the rectangle :
➠ Perimeter = 2 (Length + Breadth)
➠ 2 (36cm + 15cm)
➠ 2 (51cm)
➠ Perimeter = 102cm
Finally, finding the diagonal BD and AC of the rectangle :
➠ Diagonal = √Length² + Breadth²
➠ √(36cm)² + (15cm)²
➠ √1296 + 225
➠ √1521
➠ Diagonal = BD = AC = 39cm
(A) One of the sides of the rectangle is 15cm long.
True, BC = DA = 15cm
(B) Each diagonal of the rectangle is 38cm long.
False, the diagonals are 39cm each.
(C) Perimeter of the rectangle is 100cm.
False, the perimeter of the rectangle is 102cm.
(D) Area of the rectangle is 560cm²
False, area of the rectangle is 540cm².
Answer:—
- (A) one of the sides of the rectangle is 15 cm long.
Step by Step Explanation:—
→ AD = BC
→ AB = DC
→ 3x + 2y − 11 = x + y + 4
→ 5x + 2y + 2 = 2x + 5y − 7
→ 2x + y = 15 __________eq(1) multiplied by 3.
→ 3x−3y=−9 _______________eq(2)
on adding both equations we get.
6x + 3y = 45
3x − 3y = −9
_____________________
9x = 36
x = 4
Puting x = 4 in eq(1)
→ 2 × 4 + y = 15
→ 8 + y = 15
→ y = 15 - 8
→ y = 7
AB = 5x + 2y + 2
= 5 × 4 + 2 × 7 + 2
= 20 + 14 + 2
= 36
BC = x + y + 4
= 4 + 7 + 4
= 15
So,
Perimetre =2(36 + 15)
= 102
Area = 36 × 15 = 540
Diagonal = √15² + 36²
= √1521
= 39
@vikramjeeth.