Question

Calculate AC. Round to the nearest hundredth.

Answer 1

Length of AC is 13.19 cm

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,
`\cos x=(Base)/(Hypotenuse)`

i.e.
`\cos 65=(AD)/(7)`

i.e
`AD=7* \cos 65`

i.e
`AD=7* 0.4226`

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,
`hypotenuse^(2)=perpendicular^(2)+base^(2)`

i.e.
`AB^(2)=BD^(2)+AD^(2)`

i.e.
`BD^(2)=AB^(2)-AD^(2)`

i.e.
`BD^(2)=7^(2)-3^(2)`

i.e.
`BD^(2)=49-9`

i.e.
`BD^(2)=40`

i.e.
`BD=\pm 6.33`

Since, length of a side cannot be negative.

So, BD = 6.33 cm

Again using Pythagoras Theorem for the right triangle BDC, we have,

`BC^(2)=BD^(2)+DC^(2)`

i.e.
`BC^(2)-BD^(2)=DC^(2)`

i.e.
`12^(2)-6.33^(2)=DC^(2)`

i.e.
`DC^(2)=144-40.07`

i.e.
`DC^(2)=103.93`

i.e.
`DC=\pm 10.194`

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.