Solve the equation 9-7x-7-7x=4
1 answer:
You might be interested in
Answer:
A: The transformations are a rotation and a translation.
Step-by-step explanation:
The function f(x) is shown in the attachment as a dashed red line. The function g(x) is shown as the blue line.
The two lines have different slopes. The change in slope can be accomplished by horizontal or vertical scaling, or by rotation. The offered answer choices only include <em>rotation</em> as an option for changing the slope.
The two lines have different intercepts. To move the line from one intercept to another, generally translation is involved.
The simplest transformation is rotation and translation.
_____
<em>Comment on the answer choices</em>
If the reflection is done across a line other than an axis, or if the rotation is done about a point other than the origin, then <em>any combination of rotation, reflection, and translation can be used to make the transformation seen</em>. (Translation alone is not sufficient.)
For example, reflecting the line across the bisector of the angle between the lines (a line with a slope of (√13 -2)/3 through (5, 5)) will give the desired transformation in one step.
Similarly, a rotation of about 33.69° about the point (5, 5) will give the transformation in on step. (That angle is 45°-arctan(1/5).)
Answer:
Step-by-step explanation:
That shaded area is called a segment. To find the area of a segment within a circle, you first have to find the area of the pizza-shaped portion (called the sector), then subtract from it the area of the triangle (the sector without the shaded area forms a triangle as you can see). This difference will be the area of the segment.
The formula for the area of a sector of a circle is:
where our theta is the central angle of the circle (60 degrees) and r is the radius (the square root of 3).
Filling in:
which simplifies a bit to
which simplifies a bit further to
which of course is the same as
Now for tricky part...the area of the triangle.
We see that the central angle is 60 degrees. We also know, by the definition of a radius of a circle, that 2 of the sides of the triangle (formed by 2 radii of the circle) measure √3. If we pull that triangle out and set it to the side to work on it with the central angle at the top, we have an equilateral triangle. This is because of the Isosceles Triangle Theorem that says that if 2 sides of a triangle are congruent then the angles opposite those sides are also congruent. If the vertex angle (the angle at the top) is 60, then by the Triangle Angle-Sum theorem,
180 - 60 = 120, AND since the 2 other angles in the triangle are congruent by the Isosceles Triangle Theorem, they have to split that 120 evenly in order to be congruent. 120 / 2 = 60. This is a 60-60-60 triangle.
If we take that extracted equilateral triangle and split it straight down the middle from the vertex angle, we get a right triangle with the vertex angle half of what it was. It was 60, now it's 30. The base angles are now 90 and 60. The hypotenuse of this right triangle is the same as the radius of the circle, and the base of this right triangle is . Remember that when we split that 60-60-60 triangle down the center we split the vertex angle in half but we also split the base in half.
Using Pythagorean's Theorem we can find the height of the triangle to fill in the area formula for a triangle which is
. There are other triangle area formulas but this is the only one that gives us the correct notation of the area so it matches one of your choices.
Finding the height value using Pythagorean's Theorem:
which simplifies to
and
and
and
Taking the square root of both the 9 and the 4 (which are both perfect squares, thankfully!), we get that the height is 3/2. Now we can finally fill in the area formula for the triangle!
which simplifies to
Therefore, the area in terms of pi for that little segment is
, choice A.