Answer:
Part 1)
Or simplified:
Part 2)
The value of x for which the given expression will be the lowest is:
And the magnitude of ∠BAC is 60°.
Explanation:
We are given a ΔABC with an area of one. We are also given that AB = 2, BC = a, and CA = b. CD is a perpendicular line from C to AB.
Please refer to the diagram below.
Part 1)
Since we know that the area of the triangle is one:
Simplify:
From the Pythagorean Theorem:
Substitute:
BD will simply be (2 - x). From the Pythagorean Theorem:
Substitute:
We have the expression:
Substitute:
Part 2)
We can simplify the expression. Expand and distribute:
Simplify:
Simplify:
Since this is a quadratic with a positive leading coefficient, it will have a minimum value. Recall that the minimum value of a quadratic always occur at its vertex. The vertex is given by the formulas:
In this case, a = 2√3, b = -4, and c = (4 + 2√3).
Therefore, the x-coordinate of the vertex is:
Hence, the value of x at which our expression will be the lowest is at √3/3.
To find ∠BAC, we can use the tangent ratio. Recall that:
Substitute:
Substitute:
Therefore: