Which is the better method for perimeter of a G&T question...?

Question

This is grade 11 Geometry and Trigonometry (I know it's an easy question). My question is: How do I find the perimeter in (cm) for this question (best technique)

Question: The length of two sides of a triangle are 9.3cm and 10.5cm are the included angle is 71°. Claculate: the perimeter of the triangle

My steps:
1. draw triangle and make it into 2, right-angled triangles (the right angles are on the 10.5cm horizontal.

2. 71°+90°=161°
180°-161°=19°
The last angle of one of the right-angled triangles which has the 71° angle is 19°.

3. sin 71°=O/9.3
O=9.3(sin 71°)
=8.793322753
=8.8cm (to 1.d.p)

4. cos 71°=A/9.3
A=9.3(cos 71°)
=3.027783836
=3.0cm

5.This is for the 2nd right angled triangle:
tan x°=8.8/7.5
=1.173333333
therefore x°=49.5595008
=50°

5. cos 50°=7.5/H
H=7.5/cos 50°
=11.6679287
=11.7cm

P=11.7+10.5+9.3=31.5cm
or:

a^2+b^2=c^2
8.8^2+7.5^2=133.69
sqr133.69=11.5624
=11.6cm

P=11.6+9.3+10.5=31.4cm

Note: Which method is better? Thanks.

Is it ok to use rounded values with the: a^2+b^2=c^2
therfore c=...?

Would I get the question wrong if I put 31.4cm instead of 31.5cm if I used a different technique? Which is a better technique? Using the angle to find the last side length and then adding the two known (ones I'm given-10.5 and 9.3cm) which =31.5cm for the perimeter or using:

a^2+b^2=c^2
Perimeter=31.4cm. Please help me. Thankyou.

Answer 1

I find that the third side is 11.5 cm long, to three justifiable figures, so that the perimeter is 9.3 + 10.5 + 11.5 = 31.3 cm.

[By carrying more figures, I get an answer slightly different from either of yours. Since my method only involves one direct evaluation of the length of the third side, I believe it is superior.]

The cosine formula is made for this situation:

a^2 = b^2 + c^2 - 2 b c cos A, so that

a^2 = (9.3)^2 + (10.5)^2 - 2(9.3)(10.5) cos (71 deg.), i.e.

a^2 = 86.49 + 110.25 - 195.3(0.325568...) = 133.1565...

Therefore a = 11.5393..., or 11.5 to the 3 justifiable figures.

Live long and prosper.

Answer 2

The easy way to handle this is to use:
a^2 = b^2 + c^2 -2bc cos A.