Answer:
The side length of the square will be 11 cm.
Step-by-step explanation:
Tessa wants to make a bandana for her dog by folding a square cloth into a 45-45-90 triangle.
Her dog's neck has a circumference of about 32 cm. the folded bandana needs to be an extra 16 cm long so Tessa can tie it around her dog's neck.
So, the hypotenuse of the 45-45-90 triangle has a length of (32 - 16) = 16 cm.
As a 45-45-90 triangle is an isosceles right triangle with side length a cm (say), then
[tex]\sqrt{a^{2} + a^{2}} = 16[/tex]
⇒ 2a² = 16² = 256
⇒ a² = 128
⇒ a = 8√2 cm.
Therefore, the side length of the square will be 8√2 cm ≈ 11 cm. (Answer)
The side length of the square should be [tex]34cm[/tex] (to the nearest cm)
After folding the square cloth into a bandana, the diagonal has to be [tex]32cm+16cm=48cm[/tex] so that it can tie around the dog's neck.
Using Pythagoras' theorem, the relationship between the sides of the square cloth and the diagonal will be
[tex]diagonal=\sqrt{side^2+side^2}\\diagonal=side\sqrt{2}[/tex]
substituting and solving for the required side length, we get
[tex]48=side\sqrt{2}\\side=24\sqrt{2}\approx 34cm[/tex]
So, the required side length of the square is [tex]34cm[/tex] (to the nearest cm)
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