Question

A frame shop has been commissioned to frame 5​ black-and-white photos for an island resort. Each photo has a perimeter of 68 in. and a diagonal of 26 in. Find the dimensions of the photos.

Longer side is __in.
Shorter side is __in.

Answer 1

Longer side is 24 in

Shorter side is 10 in.

Explanation:

A rectangular photo has two dimensions, height h and length l. The height and the length are the two sides.

The perimeter of a rectangle is given by the following formula:

`P = 2(l + h)`

The diagonal is the hypothenuse of a rectangular triangle, with the length and height being the sides. So

`l^(2) + h^(2) = d^(2)`

In this problem, we have that:

`P = 68, d = 26`

So

`l^(2) + h^(2) = 26^(2)`

`l^(2) + h^(2) = 676`

`h^(2) = 676 - l^(2)`

`h = \sqrt{676 - l^(2)}`

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`P = 2(l + h)`

`P = 2l + 2h`

`P = 2l + 2\sqrt{676-l^(2)}`

`68 - 2l = 2\sqrt{676 - l^(2)}`

`(68 - 2l)^(2) = (2\sqrt{676 - l^(2)})^(2)`

`4l^(2) - 272l + 4624 = 4(676 - l^(2))`

`4l^(2) - 272l + 4624 = 2704 - 4l^(2)`

`8l^(2) - 272l + 1920 = 0`

The roots are l = 24 and l = 10.

For the height, we have that:

`h = \sqrt{676 - l^(2)}`

When l = 10

`h = \sqrt{676 - l^(2)} = \sqrt{676 - l^(2)} = √(576) = 24`

When l = 24

`h = \sqrt{676 - l^(2)} = \sqrt{676 - l^(2)} = √(100) = 10`

So

Longer side is 24 in

Shorter side is 10 in.

Answer 2

• longer side: 24 in
• shorter side: 10 in

Explanation:

The diagonal of 26 only has factors of 2 and 13, so offers a clue to the solution. 5-12-13 is a Pythagorean triple often used in algebra problems. Here, a picture with double those leg/hypotenuse values would have dimensions 10×24 with a diagonal of 26. The 10×24 dimensions will give a half-perimeter of 10+24 = 34, so a perimeter of 68.

The picture dimensions are 24 in long by 10 in wide.

_____

If you don't happen to make the connection between Pythagorean triples and picture dimensions, you can write and solve an equation for the dimensions.

We know the perimeter is 2 times the sum of length and width, so that sum is 34 inches. If w is the width, 34-w is the length. The diagonal is found using the Pythagorean theorem:

26^2 = w^2 +(34-w)^2 . . . . . . . . . . . . c^2 = a^2 + b^2

676 = w^2 + 1156 - 68w +w^2 . . . . . eliminate parentheses

w^2 -34w +240 = 0 . . . . . . subtract 676, divide by 2

(w -10)(w -24) = 0 . . . . . . . . factor

Solutions to this equation are ...

w = 10 or w = 24

One of these is the length; the other is the width.

The longer side is 24 inches; the shorter side is 10 inches.