Polygons and Circles - Quantitative Reasoning

Find the number of diagonals in a nonagon.

$27$ . For $n=9$ , diagonals are $\frac{9(6)}{2}=27$ in a nonagon.
Find the area of a sector when $r=10$ and $\theta=90$ degrees.

$25\pi$ . Sector area is $\frac{90}{360} \times pi \times 10^2 = 25pi$ .
Find each exterior angle of a regular decagon.

$36$ . For $n=10$ , each exterior angle is $360/10=36$ degrees in a regular decagon.
What is the relationship between a central angle and its intercepted arc measure (degrees)?

They are equal in degree measure. The central angle subtends the arc directly at the center, matching its measure.
State the Pythagorean theorem for a right triangle with legs $a,b$ and hypotenuse $c$ .

$a^2+b^2=c^2$ . Relates the sides of a right triangle where the square of the hypotenuse equals the sum of squares of the legs.
Find the circle equation with center $(2,-3)$ and radius $5$ .

$(x-2)^2+(y+3)^2=25$ . Standard form with center $(2,-3)$ shifts $x-2$ and $y+3$ , radius squared as $25$ .
What is the measure of an inscribed angle that intercepts an arc of measure $x$ degrees?

$\frac{x}{2}$ . Inscribed angles subtend the arc from the circumference, measuring half the arc.
What is the measure of an angle formed by a tangent and a chord intercepting arc $x$ degrees?

$\frac{x}{2}$ . Such angles measure half the intercepted arc per the tangent-chord theorem.
Identify the relationship between a radius and a tangent at the point of tangency.

They are perpendicular. The tangent is perpendicular to the radius at the contact point due to the circle's symmetry.
Identify the number of diagonals in a convex $n$ -gon.

$\frac{n(n-3)}{2}$ . Counts lines connecting non-adjacent vertices, subtracting sides and adjusting for double-counting.
State the area of a sector for central angle $\theta$ in degrees and radius $r$ .

$\frac{\theta}{360}\cdot \pi r^2$ . Proportion of the full circle area based on the central angle fraction.
What is the equation of a circle with center $(h,k)$ and radius $r$ ?

$(x-h)^2+(y-k)^2=r^2$ . Defines all points at distance $r$ from the center using squared distances.
What is each interior angle of a regular $n$ -gon?

$\frac{180(n-2)}{n}$ . Obtained by dividing the sum of interior angles by $n$ for equal distribution in a regular polygon.
Find each interior angle of a regular hexagon.

$120$ . For $n=6$ , each angle is $\frac{180(4)}{6}=120$ degrees in a regular hexagon.
What is the area of a circle with radius $r$ ?

$\pi r^2$ . Represents the integral of infinitesimal areas within the circle's boundary.
What is each exterior angle of a regular $n$ -gon?

$\frac{360}{n}$ . Calculated by dividing the total sum of exterior angles by $n$ for regularity.
State the arc length of a circle for central angle $\theta$ in degrees and radius $r$ .

$\frac{\theta}{360}\cdot 2\pi r$ . Proportion of the full circumference based on the central angle fraction.
What is the area of a triangle with base $b$ and height $h$ ?

$\frac{1}{2}bh$ . Computes the area as half the product of base and corresponding height.
What is the sum of exterior angles of any convex polygon (one per vertex)?

$360$ . Results from the fact that exterior angles sum to a full rotation around a point.
What is the area of a regular polygon with perimeter $P$ and apothem $a$ ?

$\frac{1}{2}aP$ . Represents the sum of areas of triangles formed by the apothem and each side.
State the sum of interior angles of an $n$ -gon.

$180(n-2)$ . Derived by dividing the polygon into $n-2$ triangles, each contributing $180$ degrees.
What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ ?

$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ . Applies the Pythagorean theorem in two dimensions to find straight-line distance.
What is the circumference of a circle with radius $r$ ?

$2\pi r$ . Measures the perimeter as twice pi times the radius.
Find the arc length when $r=6$ and $\theta=60$ degrees.

$2\pi$ . Arc length is $\frac{60}{360} \times 2pi \times 6 = 2pi$ .

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