Question

Assume that the wooden triangle shown is a right triangle.

​​ a. Write an equation using the Pythagorean Theorem and the measurements provided in the diagram. Hint: (leg 1)2 + (leg 2)2 = (hypotenuse)2
b. Transform each side of the equation to determine if it is an identity.

Answer 1

b.
`\displaystyle 225y^2 + 150xy + 100x^2 = 225y^2 + 150xy + 100x^2`

a.
`\displaystyle [8x + 12y]^2 + [6x + 9y]^2 = [10x + 15y]^2`

Explanation:

b.
`\displaystyle 225y^2 + 150xy + 100x^2 = 225y^2 + 150xy + 100x^2`

a.
`\displaystyle [8x + 12y]^2 + [6x + 9y]^2 = [10x + 15y]^2`

The two expressions are identical on each side of the equivalence symbol, therefore they are an identity.

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Answer 2

Part 1)
`(10x+15y)^(2)=(6x+9y)^(2)+(8x+12y)^(2)`

Part 2) The answer in the procedure

Explanation:

Part 1)

we know that

Applying the Pythagoras Theorem

`c^(2)=a^(2)+b^(2)`

we have

`c=(10x+15y)`

`a=(6x+9y)`

`b=(8x+12y)`

substitute the values

`(10x+15y)^(2)=(6x+9y)^(2)+(8x+12y)^(2)`

Part 2) Transform each side of the equation to determine if it is an identity

`(10x+15y)^(2)=(6x+9y)^(2)+(8x+12y)^(2)\\ \\100x^(2)+150xy+225y^(2)=36x^(2)+54xy+81y^(2)+64x^(2)+96xy+144y^(2)\\ \\100x^(2)+150xy+225y^(2)=100x^(2)+150xy+225y^(2)`

The left side is equal to the right side

therefore

Is an identity