How do I find A^2 and B^2 and C^2 of a triangle? Like the Pythagorean theorem but for a b and c.

i)

The question is not clear as to what are A2,B2 and C2.Assuming that normally capital letters are used for naming the angles and vertices, a solution is given.

Normally a triangle is said to be given if a picture of the triangle is given and we can measure its sides AB,BC and CA and angles A,B and C by using a scale and protractor.

If two sides and an angle of a triangle is given, then the triangle is possible to draw. In this case we have to find the remaing side and the two angles, which could be measured after construction of  the triangle . Or else, use  the formula of trigonometry like: b^2 = a^2+c^2-2ac*cosB. But this may be beyond the 9th grade syllabus. The formula, of course is cyclic, in sides a,b,c, and angles, A,B and C.First find the remaing unknown side b. Then find the remaining angles using the sane formula or sine rule formula.

The triangle may be constructed with a side and two angles at the end of the given side if the data is in that form. After construction the measurements of the remaining angle and sides could be measured. Or else, find the third angle using the fact that sum of the three angles in a triangle is 180 degree. You can also use trigonometry : a/sinA=b/sinB=c/sinC, where a, b, c are the opposite sides of angles A,B and C respectively.

If all three sides are are given, then only angles are to be determined. This could be measured  by constructing the triangle. Or use a formula like: a^2=b^2+c^2-2bc* cosA, and determine the angle A. Similarly you can find other angles cylically changing the formula.

ii)

This  is a relation between angles in a right angled triangle, which is similar to pytagorus formula.

In a right angled triangle ABC with right angle at B,

AB^2+BC^2=AC^2, by Pythagorus therem.

Divide AC^2 both sides:

AB^2/AC^2+BC^2/AC^2= 1............................(1)

By trigonometry, AB/AC=sinC,  BC/AC=sinA , and 1=sinB.

Therefore, the equation at (1) due to Pythagorus could be written like: (Sin C)^2+(Sin A)^2 = (sin B)^2.

Therefore,

A^2 = [arc sin (BC/AC)]^2

B^2 = 1^2 =1 as angle B = 90deg.

C^2= {Arc sin (AB/AC)}^2.

Hope this may help.

iii)

If you want to know how to use the formula of Pythagorus:

Example 1:

Given the the two sides of a right angled triangle to find the third side: 3cm and 4cm are sides making right angled triangle. Find the 3rd side.

a^2+c^2=b^2

3^2+4^2=b^2. or b^2=25 or b=sqrt25=5cm

Example 2:

Given the hypotenuse = 17cm and one side 15cm. Find the other side of the right angled triangle:

a^2+b^2=c^2

a^2+15^2=17^2

a^2=17^2-15^2= 289-225=64

Therefore a=sqrt (64)=8cm