Answer: C) ASA = ASA
Refer to the diagram below. I've used color coding to match up the given statements with the drawing. Note how the red segments are between the blue and green angles. So we use ASA to prove the triangles are congruent.
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Answer:
(√5)/2
Step-by-step explanation:
Of the several ways I can think of to do this, using a graphing calculator is about the easiest. It shows the minimum to be ...
f(1) = √1.25 = (√5)/2
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Using the distance formula, you have ...
f(x) = √((x -(-2))² +((x² +2)-5/2)²)
f(x) = √(x² +4x +4 +x⁴ -x² +1/4) = √(x⁴ +4x +17/4)
The minimum is found where the derivative is zero.
f'(x) = (2x³ +2)/√(x⁴ +4x +17/4) = 0
x³ = -1 . . . . . f'(x) is zero when the numerator is zero
x = -1 . . . . . cube root
Then the minimum value of f(x) is ...
f(-1) = √(x⁴ +4x +17/4) = √((-1)⁴ +4(-1) +17/4) = √(1 -4 +17/4) = √(5/4)
f(-1) = (√5)/2 . . . . minimum value of f(x)
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The graph shows f²(x) in red and its minimum of 1.25 = 5/4. The curve (x, x²+2) and the point (-2, 5/2) are also shown, for reference. (The slope of the curve at x=-1 is -2, and the normal to the curve at that point has slope 1/2. The normal goes through the point (-2, 5/2), consistent with f(x) being a minimum at x=-1.)
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