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.
Accepted Solution
A:
Answer:longer side: 24 inshorter side: 10 inStep-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 . . . . . . . . factorSolutions to this equation are ... w = 10 or w = 24One of these is the length; the other is the width.The longer side is 24 inches; the shorter side is 10 inches.