The triangle ABD with the title 1:AB = 10, BD=6 and AD=8 is a right triangle (according to Pythagorean theorem).

Advertising vertical BC

So the triangle ACD is a right triangle.

So according to Pythagorean theorem, DC= 15.

Because AC = 17, AB = 10, BC = BD+DC = 6+ 15 = 2 1, Pythagorean theorem does not hold.

So triangle ABC is not a right triangle.

The area of ABC =BC*AD/2=2 1*8/2=84.

Topic 2

Because the angle ACB=90 degrees, BC = 3, AC = 4.

So according to Pythagorean theorem, AB=5.

According to the triangle area formula, AB*CD/2=BC*AC/2.

Get CD= 12/5

In right triangle BCD, BD=9/5 according to Pythagorean theorem.

Theme 3

Party A+Party B+Party C +200= 12a+ 16b+20c

(Party A-12a+36)+ (Party B-16b+64)+ (Party C -20c+ 100)=0.

(Party A-12a+6)+ (Party B-16b+8)+ (Party C -20c+ 10) =0.

(a-6) party +(b-8) party +(c- 10) party =0.

So (a-6) square = 0; (b-8) Square = 0; (c- 10) squared =0.

So a = 6；; b = 8； c= 10

So the triangle ABC is a right triangle (6 +8 = 10).