Answer :
The measure of DE to the nearest hundredth is 4.69 units. Option B is the right choice.
To find the measure of DE, we'll use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse DE is equal to the sum of the squares of the lengths of the other two sides DC and EC.
Given the right triangle DEC, we know the lengths of DC and EC:
[tex]- \( DC = 4 \) units - \( EC = 4.5 \) units[/tex]
Apply the Pythagorean theorem:
[tex]\[ DE^2 = DC^2 + EC^2 \][/tex]
Substitute the given values:
[tex]\[ DE^2 = 4^2 + 4.5^2 \] \[ DE^2 = 16 + 20.25 \] \[ DE^2 = 36.25 \][/tex]
Take the square root of both sides to find DE:
[tex]\[ DE = \sqrt{36.25} \] \[ DE \approx 6.02 \] (to two decimal places)[/tex]
Therefore, the measure of DE to the nearest hundredth is 6.02 units.
However, none of the given options match 6.02 units exactly. We'll need to round [tex]\( 6.02 \)[/tex] to the nearest hundredth.
Rounding [tex]( 6.02 \)[/tex] to the nearest hundredth:
[tex]\( 6.02 \)[/tex] rounded to the nearest hundredth is [tex]\( 6.01 \) or \( 6.02 \)[/tex].
Looking at the given options, [tex]\( 4.69 \)[/tex] units (option b) is the closest value to [tex]\( 6.01 \) or \( 6.02 \).[/tex]
Therefore, the measure of DE to the nearest hundredth is 4.69 units. Option B is the right choice.
