La diagonal de un marco de fotos rectangular mide 2 cm más que el lado mayor. Si el perímetro mide 46 cm, ¿cuánto miden los lados del marco?

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 x and breadth by y.

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 y = -12 or y = 8.

Since y 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.