Definition of rectangle :
A rectangle is a parallelogram with four equal angles of 90 degree.
Properties of rectangle :
Problem 1 :
In the given rectangle, find ∠1 and ∠2.
Solution :
∠PMQ + ∠MQP + ∠QPM = 180
3x + 90 + 2x + 20 = 180
5x + 110 = 180
Subtracting 110 on both sides.
5x = 180 - 110
5x = 70
Dividing by 5 on both sides.
x = 70/5
x = 14
∠2 = ∠PMQ ∠PMQ = 3x = 3(14) ∠PMQ = 42 |
∠1 = ∠QPM ∠QPM = 2x + 20 = 2(14) + 20 ∠QPM = 48 |
Problem 2 :
Find x.
Solution :
Since the diagonals are equal, the value of x is 14 cm.
Problem 3 :
In the rectangle given below, find the value of x.
Solution :
Since the diagonals are equal and bisect each other,
OA = OD
OA is half of diagonal AC and OD is half of the diagonal of BD.
2x + 4 = 3x + 1
Subtracting 3x and 4 on both sides.
2x - 3x = 1 - 4
-x = -3
x = 3
Problem 4 :
In the rectangle given below, the length of the diagonal is 36. Find values of x and y.
Solution :
Length of diagonal = 36
2x + 4y + 4x - y = 36
6x + 3y = 36
Dividing by 2, we get
2x + y = 12 ----(1)
Length of diagonal will be equal.
2x + 4y = 4x - y
2x - 4x + 4y + y = 0
-2x + 5y = 0 ----(2)
(1) + (2)
2x - 2x + y + 5y = 12 + 0
6y = 12
y = 12/6
y = 2
By applying the value of y in (1), we get
2x + 2 = 12
2x = 10
x = 10/2
x = 5
Problem 5 :
In rectangle ABCD, AC = x + 8, AB = x + 7 and AD = x. Find the value of x, also find AC, AB and AD.
Solution :
In triangle ABC, angle B is 90 degree.
AC^{2} = AB^{2} + BC^{2}
(x + 8)^{2} = (x + 7)^{2} + x^{2}
x^{2} + 16x + 64 = x^{2} + 14x + 49 + x^{2}
2x^{2} - x^{2} + 14x - 16x + 49 - 64 = 0
x^{2} - 2x - 15 = 0
(x - 5) (x + 3) = 0
x = 5 and x = -3
AB = x + 7 If x = 5 AB = 12 If x = -3 AC = 4 |
AC = x + 8 If x = 5 AC = 13 If x = -3 AC = 5 |
AD = x AD = 5 AD = -3 (not acceptable) |
So, AB = 12, AC = 13 and AD = 5.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM