Answer:
Area = 78.98 square inches
Explanation:
We can use the formula for the area of a triangle, which is:
Area = (1/2) * base * height
where the base and height are both unknown. To find them, we can use the law of sines, which states that for any triangle ABC:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, and c are the side lengths and A, B, and C are the angles opposite those sides.
In this case, we know that AC = 10, BC = 2.5, and C = 63°. Let's call AB = c, and let h be the height from C to AB. Then we have:
10/sin(A) = c/sin(B)
2.5/sin(A) = c/sin(180° - B - A) = c/sin(C)
Using the first two equations, we can solve for c and sin(A) in terms of sin(B):
c = (10sin(B))/sin(A)
2.5 = (csin(A))/sin(B)
= (10*sin(A))/sin(C)
Substituting c from the first equation into the second equation, we get:
2.5 = (10*sin(A)^2)/(sin(B)*sin(C))
Solving for sin(A), we get:
sin(A) = sqrt((2.5*sin(B)sin(C))/10)
= sqrt(0.157sin(B)*sin(C))
Now we can use the formula for the area of a triangle:
Area = (1/2) * AB * h
where h is the height from C to AB. We can find h using sin(A):
h = ACsin(A) = 10sin(A)
Substituting the expression for sin(A), we get:
h = 10sqrt(0.157sin(B)*sin(C))
Finally, we can find the area:
Area = (1/2) * AB * h = (1/2) * c * h = (1/2) * (10sin(B))/sin(A) * 10sqrt(0.157*sin(B)*sin(C))
Area = 78.98 square inches
Therefore, the area of △ABC is approximately 78.98 square inches, rounded to the nearest hundredth.