Answer:
Length of AC is 13.19 cm
Step-by-step explanation:
We have the right triangle ADB with an angle 65° and the length of hypotenuse = 7 cm.
As we know, 'In a right angled triangle, the angles and sides can be written in trigonometric forms'.
That is, [tex]\cos x=\frac{Base}{Hypotenuse}[/tex]
i.e. [tex]\cos 65=\frac{AD}{7}[/tex]
i.e [tex]AD=7\times \cos 65[/tex]
i.e [tex]AD=7\times 0.4226[/tex]
i.e. AD = 3 cm
Also, Pythagoras Theorem' states that 'The sum of squares of the length of the sides in a right triangle is equal to the square of the length of the hypotenuse'.
That is, [tex]hypotenuse^{2}=perpendicular^{2}+base^{2}[/tex]
i.e. [tex]AB^{2}=BD^{2}+AD^{2}[/tex]
i.e. [tex]BD^{2}=AB^{2}-AD^{2}[/tex]
i.e. [tex]BD^{2}=7^{2}-3^{2}[/tex]
i.e. [tex]BD^{2}=49-9[/tex]
i.e. [tex]BD^{2}=40[/tex]
i.e. [tex]BD=\pm 6.33[/tex]
Since, length of a side cannot be negative.
So, BD = 6.33 cm
Again using Pythagoras Theorem for the right triangle BDC, we have,
[tex]BC^{2}=BD^{2}+DC^{2}[/tex]
i.e. [tex]BC^{2}-BD^{2}=DC^{2}[/tex]
i.e. [tex]12^{2}-6.33^{2}=DC^{2}[/tex]
i.e. [tex]DC^{2}=144-40.07[/tex]
i.e. [tex]DC^{2}=103.93[/tex]
i.e. [tex]DC=\pm 10.194[/tex]
Since, length of a side cannot be negative.
So, DC = 10.194 cm.
Finally, as the side AC is the sum of segments AD and DC, we have,
AC = AD + DC
i.e. AC = 3 + 10.194
i.e. AC = 13.19 cm
Hence, the length of AC is 13.19 cm.
