Question

triangle def is shown below.
what is the length, in centimeters, of \\(\overline{ef}\\)?

Explanation:

Step1: Identify the triangle type

Triangle \( DEF \) is a right - triangle with \( \angle E = 90^{\circ} \), \( DE = 6\space cm \) and \( DF=10\space cm \). We can use the Pythagorean theorem which states that in a right - triangle \( a^{2}+b^{2}=c^{2} \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs of the right - triangle. Let \( EF = x \), \( DE = 6 \) and \( DF = 10 \). Here, \( DF \) is the hypotenuse, \( DE \) and \( EF \) are the legs. So by the Pythagorean theorem, \( DE^{2}+EF^{2}=DF^{2} \).

Step2: Substitute the known values

Substitute \( DE = 6 \) and \( DF = 10 \) into the Pythagorean theorem formula:
\( 6^{2}+x^{2}=10^{2} \)
We know that \( 6^{2}=36 \) and \( 10^{2} = 100 \), so the equation becomes:
\( 36+x^{2}=100 \)

Step3: Solve for \( x \)

Subtract 36 from both sides of the equation:
\( x^{2}=100 - 36 \)
\( x^{2}=64 \)
Take the square root of both sides. Since \( x \) represents the length of a side of a triangle, we take the positive square root. So \( x=\sqrt{64} = 8 \)

Answer:

\( 8 \)