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2 years ago

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Points A and B lie on a circle centered at point O. If OA=5 and lenth of AB/circumference=1/4 what is the area of sector AOB? Us

e the value pi =3.14

1 answer:

MakcuM [25]2 years ago

4 0

<h3>Given</h3>

arc AB of circle O is 1/4 of the circumference
radius of circle O is 5
pi is 3.14

<h3>Find</h3>

area of sector AOB

<h3>Solution</h3>

Since arc AB is 1/4 of the circumference, the central angle AOB will be 1/4 of a circle, so π/2 in radians. The area of a sector is given by

... A = (1/2)r²·θ

where θ is the central angle in radians and r is the radus.

The area is ...

... A = (1/2)·5²·π/2 = 25π/4

... A ≈ 19.625

The area of the sector is about 19.63 square units.

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In the image attached you can find the Unit 7 homework.

We need to findt he missing measures of each figure.

<h3>1.</h3>

Notice that the first figure is a rectangle, which means opposite sides are congruent so,

VY = 19

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VW = 31

To find the diagonals we need to use Pythagorean's Theorem, where the diagonals are hypothenuses.

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That means, $ZX = \frac{VX}{2} \approx \frac{36.36}{2}\approx 18.18$

<h3>2.</h3>

Figure number two is also a rectangle.

If GH = 14, that means diagonal GE = 28, because diagonals intersect in equal parts.

Now, GF = 11, because rectangles have opposite sides congruent.

DF = 28, because in a reactangle, diagonals are congruent.

HF = 14, because its half of a diagonal.

To find side DG, we need to use Pythagorean's Theorem, where GE is hypothenuse

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<h3>3.</h3>

This figure is also a rectangle, which means all four interior angles are right, that is, equal to 90°, which means angle 11 and the 59° angle are complementary, so

$\angle 11 +59\°=90\°\\\angle 11=90\°-59\°\\\angle 11=31\°$

Now, angles 11 and 4 are alternate interior angles which are congruent, because a rectangle has opposite congruent and parallel sides.

$\angle 4 = 31\°$

Which means $\angle 3 = 59\°$ , beacuse it's the complement for angle 4.

Now, $\angle 6 = 59\°$ , because it's a base angle of a isosceles triangle. Remember that in a rectangle, diagonals are congruent, and they intersect equally, which creates isosceles triangles.

$\angle 9=180-59-59=62$ , by interior angles theorem.

$\angle 8 =62$ , by vertical angles theorem.

$\angle 10 = 180- \angle 9=180-62=118\°$ , by supplementary angles.

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$\angle 2 = \angle 5 = 31\°$ , by alternate interior angles.

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$m\angle ADE=90-6=84$ , by complementary angles.

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$m\angle DEA=180-168=12$ , by supplementary angles.

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$m\angle JMK=180-126=54$ , by supplementary angles.

$m\angle JKH=\frac{180-54}{2}=\frac{126}{2}=63$ , by interior angles theorem, and by isosceles triangle theorem.

$m\angle HLK=90\°$ , by definition of rectangle.

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$m\angle LHK=90-27=63$ , by complementary angles.

$m\angle = JLK= m\angle HJL=27$ , by alternate interior angles.

<h3>6.</h3>

The figure is a rectangle, which means its opposite sides are equal, so

$WZ=XY\\7x-6=3x+14\\7x-3x=14+6\\4x=20\\x=\frac{20}{4}\\ x=5$

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Then,

$PR=5x+28=5(9)+28=45+28=73$

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