Mathematics, 21.10.2019 13:20 ceejay8005

Aframe 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.

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Answer from: qveenriles3714

longer side: 24 inshorter side: 10 in

Step-by-step 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.

Answer from: withrowrj

Longer side is 24 in

Shorter side is 10 in.

Step-by-step 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

$l^{2} + h^{2} = 26^{2}$

$l^{2} + h^{2} = 676$

$h^{2} = 676 - l^{2}$

$h = \sqrt{676 - l^{2}}$

-------

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

$h = \sqrt{676 - l^{2}} = \sqrt{676 - l^{2}} = \sqrt{576} = 24$

When l = 24

$h = \sqrt{676 - l^{2}} = \sqrt{676 - l^{2}} = \sqrt{100} = 10$

Longer side is 24 in

Shorter side is 10 in.

Answer from: Quest

answer: 590

step-by-step explanation:

Answer from: Quest

each side has a different length.

step-by-step explanation:

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Aframe shop has been commissioned to frame 5 black-and-white photos for an island resort. each photo...