Respuesta :
Answer:
- longer side: 24 in
- shorter 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.
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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:
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:
[tex]P = 2(l + h)[/tex]
The diagonal is the hypothenuse of a rectangular triangle, with the length and height being the sides. So
[tex]l^{2} + h^{2} = d^{2}[/tex]
In this problem, we have that:
[tex]P = 68, d = 26[/tex]
So
[tex]l^{2} + h^{2} = 26^{2}[/tex]
[tex]l^{2} + h^{2} = 676[/tex]
[tex]h^{2} = 676 - l^{2}[/tex]
[tex]h = \sqrt{676 - l^{2}}[/tex]
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[tex]P = 2(l + h)[/tex]
[tex]P = 2l + 2h[/tex]
[tex]P = 2l + 2\sqrt{676-l^{2}}[/tex]
[tex]68 - 2l = 2\sqrt{676 - l^{2}}[/tex]
[tex](68 - 2l)^{2} = (2\sqrt{676 - l^{2}})^{2}[/tex]
[tex]4l^{2} - 272l + 4624 = 4(676 - l^{2})[/tex]
[tex]4l^{2} - 272l + 4624 = 2704 - 4l^{2}[/tex]
[tex]8l^{2} - 272l + 1920 = 0[/tex]
The roots are l = 24 and l = 10.
For the height, we have that:
[tex]h = \sqrt{676 - l^{2}}[/tex]
When l = 10
[tex]h = \sqrt{676 - l^{2}} = \sqrt{676 - l^{2}} = \sqrt{576} = 24[/tex]
When l = 24
[tex]h = \sqrt{676 - l^{2}} = \sqrt{676 - l^{2}} = \sqrt{100} = 10[/tex]
So
Longer side is 24 in
Shorter side is 10 in.
