Question

Rob wants to rebuild the frame for the roof of his house, so he made a model of the frame first. The width of the frame is 6 feet and one of its sides is 5 feet. If the height of the frame is 3 feet, what is the length of the other side of the frame?

• A. $\sqrt{15}ft$
• B. $\sqrt{13}ft$
• C. $\sqrt{19}ft$
• D. $\sqrt{17}ft$

Answer 1

The correct option is B $\sqrt{13}ft$

One side length of the frame = 5 ft
Width of the frame = 6 ft
Height of the frame = 3 ft
The triangle formed is shown below:

In $\Delta CDB$,
$B{C}^2=C{D}^2+D{B}^2$ (by Pythagorean theorem)
$\Rightarrow 5^2=3^2+D{B}^2$ (substitute 5 for BC and 3 for CD)
$\Rightarrow 25=9+D{B}^2$
$\Rightarrow D{B}^2=25-9$
$\Rightarrow D{B}^2=16$
$\Rightarrow DB=\sqrt{16}$ (evaluate the powers)
$\Rightarrow DB=4$ (take the positive square for each side)
We also know that
$AD+DB=6$.
Hence, $AD=6-4=2ft.$

In $\Delta CDA$,
$A{C}^2=A{D}^2+D{C}^2$ (by Pythagorean theorem)
$\Rightarrow x^2=3^2+2^2$ (substitute 3 for AD and 2 for DC)
$\Rightarrow x^2=9+4$
$\Rightarrow x^2=13$
$\Rightarrow x=\sqrt{13}$
The side length of the frame is $\sqrt{1}3ft$.
Hence, the correct answer is option A.