If the quadrilateral below is a rhombus, find the missing measures.
1 answer:
Answer:
- corner angles are 80° (∠JKL) and 100° (∠MJK)
- side lengths are 2√61
Step-by-step explanation:
The given length measures are the diagonals of the rhombus, so each side will be the hypotenuse of the triangle with legs half as long:
√(12² +10²) = √244 = 2√61 ≈ 15.62 . . . . side length
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The given angle measure is half the measure at ∠J. The full measure of ∠J is 100°, so the measure of ∠K is 180° -100° = 80°
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Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
which is a cubic polynomial in with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)