Find the area of a circle with a circumference of 53.42 inches

Mathematics

1 answer:

Sidana [21]3 years ago

3 0

Answer:

226.9 in²

Step-by-step explanation:

The formula for circumference of a circle is:

C = 2πr, where π = 3.14 and r = the radius

The formula for area of a circle is

A = πr²

First, using the formula for circumference, you can solve for 'r':

53.42 = 2πr or 53.42 = 6.28r (divide) r = 8.5

Now, use the value of 'r' to solve for area:

A = π8.5² or 3.14(8.5)² = 226.9 in²

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Knowing this, we should start by doing the equations <em>within </em>each parentheses.

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2. Given a quadrilateral with vertices (−1, 3), (1, 5), (5, 1), and (3,−1):

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<h2>Explanation:</h2>

In every rectangle, the two diagonals have the same length. If a quadrilateral's diagonals have the same length, that doesn't mean it has to be a rectangle, but if a parallelogram's diagonals have the same length, then it's definitely a rectangle.

So first of all, let's prove this is a parallelogram. The basic definition of a parallelogram is that it is a quadrilateral where both pairs of opposite sides are parallel.

So let's name the vertices as:

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First pair of opposite sides:

<u>Slope:</u>

$\text{For AB}: \\ \\ m=\frac{5-3}{1-(-1)}=1 \\ \\ \\ \text{For CD}: \\ \\ m=\frac{1-(-1)}{5-3}=1 \\ \\ \\ \text{So AB and CD are parallel}$

Second pair of opposite sides:

<u>Slope:</u>

$\text{For BC}: \\ \\ m=\frac{1-5}{5-1}=-1 \\ \\ \\ \text{For AD}: \\ \\ m=\frac{-1-3}{3-(-1)}=-1 \\ \\ \\ \text{So BC and AD are parallel}$

So in fact this is a parallelogram. The other thing we need to prove is that the diagonals measure the same. Using distance formula:

$d=\sqrt{(y_{2}-y_{1})^2+(x_{2}-x_{1})^2} \\ \\ \\ Diagonal \ BD: \\ \\ d=\sqrt{(5-(-1))^2+(1-3)^2}=2\sqrt{10} \\ \\ \\ Diagonal \ AC: \\ \\ d=\sqrt{(3-1)^2+(-5-1)^2}=2\sqrt{10} \\ \\ \\$