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katrin [286]
3 years ago
15

RATE is a quadrilateral inscribed in the above circle with arc measures shown above. What is the measure of the smallest angle i

n RATE?

Mathematics
2 answers:
Sav [38]3 years ago
8 0

Answer:

The measure of the smallest angle is 77°

Step-by-step explanation:

* Lats study some facts about the circle

* In any circle; If the vertices of a quadrilateral lie on its

 circumference then the quadrilateral is called a cyclic quadrilateral

- In any cyclic quadrilateral each two opposite angles are supplementary

 that means their sum = 180°

* In any circle; if the vertex of an angle lies on the circumference

 is called an inscribed angle

- The inscribed angle subtended by the opposite arc

- The measure of the inscribed angle = 1/2 the measure of

  the subtended arc

* Now we can solve our question

∵ R , A , T , E all on the the circumference of the circle

∴ RATE is a cyclic quadrilateral

∵ Angle RAT is an inscribed angle subtended by arc RET

∵ The measure of arc RE = 121°

∵ The measure of arc ET = 51°

∴ The measure of arc RET = 121 + 51 = 172°

∵ m∠RAT = (1/2) measure of arc RET

∴ m∠RAT = (1/2) × 172 = 86°

∵ Angle ATE is an inscribed angle subtended by arc ARE

∵ The measure of arc AR = 85°

∵ The measure of arc RE = 121°

∴ The measure of arc ARE = 85 + 121 = 206°

∵ m∠ATE = (1/2) measure of arc ARE

∴ m∠ATE = (1/2) × 206 = 103°

* Now lets find the remainder angles by using cyclic quadrilateral

∵ RATE is a cyclic quadrilateral

∴ m∠RAT + m∠RET = 180°

∵ m∠RAT = 86°

∴ m∠RET = 180 - 86 = 94°

* Similar;

∵ RATE is a cyclic quadrilateral

∴ m∠ATE + m∠ARE = 180°

∵ m∠ATE = 103°

∴ m∠ARE = 180 - 103 = 77°

* The measure of the smallest angle is 77°

Dima020 [189]3 years ago
8 0

Answer:

77

Step-by-step explanation:

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​Arif's backyard is 24 by 37 meters. He wants to put a flower garden in the middle of the​ backyard, leaving a strip of grass of
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Answer:

Length of the flower garden = 29 meters

Width of the flower garden = 16 meters

Step-by-step explanation:

I have also attached a diagram for your convenience

Now First let us see the data we are given with:

Arif's backyard length = 37 meters

Arif's backyard Width = 24 meters

Area that should be left with grass = 424 m^2

Now lets suppose the width of the stirp of grass be "a"

According the image the area of backyard will be:

Area ABCD = 24 x 37

Area ABCD = 888 m^2

The length LM will be = (37 -2a)

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Area LMNO = (37 -2a) x  (24 -2a)

Area LMNO = 888 -74a -48a +4a^2

Area LMNO = 4a^2 -122a + 888

Now in order the to calculate the area of the grass strip we must subtract the area LMNO from the total area ABCD and we already know that the areas the area of grass strip should be 424 m^2

hence, A = Area ABCD - Area LMNO

424 = 888 - (4a^2 -122a + 888)

424 = 888 - 4a^2 +122a - 888

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2(2x-53)(x-4) = 0

2(2x-53) = 0

2x -53 = 0

x = 53/2

x = 26.5 (Not Possible)

x-4 = 0

x = 4

Length of the flower garden = 37 -2(4)

Length of the flower garden = 29 meters

Width of the flower garden = 24 -2(4)

Width of the flower garden = 16 meters

4 0
3 years ago
amy is going to put 6 triangular tables together to make one large hexagon shaped table. what will be the area of the hexagon ta
tresset_1 [31]

Answer:

The area of the hexagonal table will be 27 square feet

Step-by-step explanation:

Given:

The height of the triangle = 3 feet

The base of the triangle = 3 feet

To Find:

The area of the hexagonal table = "

Solution:

<u>Step 1: Finding the area of the triangle</u>

The area of the triangle  =\frac{1}{2} \times base \times height

On substituting the values

The area of the triangle is =\frac{1}{2} \times 3 \times 3

The area of the triangle  = \frac{9}{2} square feet

<u>Step 2 : Finding the area of the hexagonal table </u>

The area of the hexagonal table is  =  6  X area of one triangle

The area of the hexagonal table  = 6 \times \frac{9}{2}

The area of the hexagonal table  = \frac{54}{2}

The area of the hexagonal table  = 27 square feet

3 0
3 years ago
Read 2 more answers
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