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Hermina cut a [tex][tex]$10^{\prime \prime}$[/tex][/tex] by [tex][tex]$20^{\prime \prime}$[/tex][/tex] piece of cardboard down the diagonal.

What is the length [tex]\( c \)[/tex] of the cut, in inches?

A. [tex][tex]$\sqrt{500}$[/tex][/tex] inches
B. 30 inches
C. [tex][tex]$\sqrt{300}$[/tex][/tex] inches
D. [tex][tex]$\sqrt{60}$[/tex][/tex] inches


Sagot :

Sure! Let's find the length of the diagonal for a rectangle with sides of 10 inches and 20 inches.

1. Identify the given sides of the rectangle:
- Length [tex]\( a = 10 \)[/tex] inches
- Width [tex]\( b = 20 \)[/tex] inches

2. Recall the Pythagorean theorem for calculating the diagonal [tex]\( c \)[/tex] of a rectangle:
The Pythagorean theorem states that for a right triangle,
[tex]\[ c^2 = a^2 + b^2 \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two shorter sides, and [tex]\( c \)[/tex] is the length of the hypotenuse (or diagonal in this case).

3. Plug in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the theorem:
[tex]\[ c^2 = (10)^2 + (20)^2 = 100 + 400 = 500 \][/tex]

4. Solve for [tex]\( c \)[/tex] by taking the square root of both sides:
[tex]\[ c = \sqrt{500} \][/tex]

5. Comparison with given options:
Now, we compare the calculated diagonal length with the provided options:

- [tex]\( \sqrt{500} \)[/tex] inches.
- 30 inches.
- [tex]\( \sqrt{300} \)[/tex] inches.
- [tex]\( \sqrt{60} \)[/tex] inches.

Among the options provided, the length of the cut, which is the diagonal of the rectangle, is [tex]\( \sqrt{500} \)[/tex] inches.

Answer:
The length of the cut is [tex]\( \sqrt{500} \)[/tex] inches.