Answer:
The length of rectangular photo frame is 15 cm and the breadth is 8 cm.
Step-by-step explanation:
The question is:
The diagonal of a rectangular photo frame is 2 cm more than the longest side. If the perimeter is 46 cm, how long are the sides of the frame?
Solution:
Let the length of the rectangular photo frame be denoted by <em>x</em> and breadth by <em>y</em>.
It is provided that the diagonal is 2 cm more than the length.
That is:
d = x + 2
The perimeter is 46 cm.
That is:
46 = 2 (x + y)
⇒ x + y = 23
⇒ x = 23 - y
The triangle form by the length, breadth and the diagonal of the rectangle is a right angled triangle, with the diagonal as the hypotenuse, length as perpendicular and breadth as the base.
So, according to the Pythagoras theorem,
d² = x² + y²
(x + 2)² = x² + y²
x² + 4x + 4 = y²
4x + 4 = y²
4 (23 - y) + 4 = y²
92 - 4y + 4 = y²
y² + 4y - 96= 0
Factorize the expression by splitting the middle term as follows:
y² + 4y - 96= 0
y² + 12y - 8y - 96= 0
y (y + 12) - 8 (y + 12) = 0
(y + 12)(y - 8) = 0
Either <em>y</em> = -12 or <em>y</em> = 8.
Since <em>y</em> represents the breadth of a rectangle, it cannot be negative.
Thus, the breadth of rectangular photo frame is 8 cm.
Compute the length as follows:
x = 23 - y
= 23 - 8
= 15
Thus, the length of rectangular photo frame is 15 cm.
