Question

Square STQR is drawn on one side of right triangle TPQ. The length of each side of the square is about units, and the length of its diagonal is about units.

Answer 1

the length of its diagonal is √2

Explanation:

Data provided:

The square is drawn on one side of the right triangle

also,

Length of each side of square is 1 units

Now, let the length of the diagonal be 'X'

also,

The diagonal of the square will be the hypotenuse of the right triangle

we know the property of the right triangle that

Base² + Perpendicular² = Hypotenuse²

OR

1² + 1² = X²

or

X² = 2

or

X = √2

Hence,

the length of its diagonal is √2

Answer 2

Each side of the square as a lenght of
`2√(5)` units.

The diagonal length is
`2√(10)` units.

Explanation:

In the image attached, you can observe that one leg of the right triangle has the same length than a side of the square below, that means we need to find that leg length.

Using Pytagorean's Theorem, we have

`6^(2)=4^(2)+l^(2) \\36-16=l^(2)\\ l=√(20)=2√(5)`

Therefore, each side of the square as a lenght of
`2√(5)` units.

Now, notice that the right triangle inside the square has its hypothesus congruent with the diagonal, using the lengt of each side of the square, we can find the diagonal length.

`d^(2)=(2√(5))^(2) +(2√(5))^(2) =4(5)+4(5)=20+20=40\\d=√(40)\\ d=2√(10)`

Therefore, the diagonal length is
`2√(10)` units.