SOLUTION: Directions state: Find the perimeters and areas of the squares and rectangles in questions 7-10. The figure shown in the text has a picture of a square which is dissected in th

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Question 198847This question is from textbook
: Directions state: Find the perimeters and areas of the squares and rectangles in questions 7-10.
The figure shown in the text has a picture of a square which is dissected in the middle to form two right triangles. The triangle has two 45 degree and one 90 degree angles. All that is given is six square root of two (6 square root 2) for the hypotenuse of the adjacent right triangles. My apoogies but I do not have a square root symbol key on my computer keyboard. As you can tell I need to solve for the perimeter and area, but I do not even know how to without data given for either side one or side two.This question is from textbook

Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
If we just focus on one triangle, we get

drawing%28500%2C500%2C-0.5%2C2%2C-0.5%2C3.2%2C%0D%0Aline%280%2C0%2C0%2C3%29%2C%0D%0Aline%280%2C3%2C2%2C0%29%2C%0D%0Aline%282%2C0%2C0%2C0%29%2C%0D%0Alocate%28-0.2%2C1.5%2Cx%29%2C%0D%0Alocate%281%2C-0.2%2Cx%29%2C%0D%0Alocate%281%2C2%2C6%2Asqrt%282%29%29%0D%0A%29


To find the unknown length, we need to use the Pythagorean Theorem.


Remember, the Pythagorean Theorem is a%5E2%2Bb%5E2=c%5E2 where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since the legs are x and x (ie the legs are equal) this means that a=x and b=x


Also, since the hypotenuse is 6%2Asqrt%282%29, this means that c=6%2Asqrt%282%29.


a%5E2%2Bb%5E2=c%5E2 Start with the Pythagorean theorem.


x%5E2%2Bx%5E2=%286%2Asqrt%282%29%29%5E2 Plug in a=x, b=2, c=6%2Asqrt%282%29


x%5E2%2Bx%5E2=72 Square 6%2Asqrt%282%29 to get %286%2Asqrt%282%29%29%5E2=%286%2Asqrt%282%29%29%286%2Asqrt%282%29%29=36%2A%28sqrt%282%29%29%5E2=36%2A2=72.


2x%5E2=72 Combine like terms.


x%5E2=72%2F2 Divide both sides by 2.


x%5E2=36 Reduce


x=sqrt%2836%29 Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


x=6 Evaluate the square root of 36 to get 6


So the side length of the square is 6 units.


Now simply plug this side length into the following formulas (note: "s" is the length of the side of the square):


Perimeter: P=4s=4%286%29=24. So the perimeter is 24 units.

Area: P=s%5E2=%286%29%5E2=36. So the area is 36 square units.

Answer by solver91311(12126) About Me  (Show Source):
You can put this solution on YOUR website!


Since it is a square, the triangle formed by two adjacent sides of the square and the diagonal is an isosceles right triangle -- observation confirmed by the fact that you say it is a 90°-45°-45° triangle. All isosceles right triangles are similar, that is to say that their sides are in proportion.

If you have an isosceles right triangle whose legs measure 1 unit, then the Pythagorean Theorem says the hypotenuse is:



That means that the three sides of any isosceles right triangle are in proportion:



So, if your hypotenuse, i.e. the diagonal of the square, measures , then the two legs must each measure 6 because



is equivalent to



Can you take it from there?

John