Mathematics : asked on Z0D

31.12.2022

Pls ! triangle xyz was dilated by a scale factor of 2 to create triangle acb and sin ∠x =5/5.59
.
part a: use complete sentences to explain the special relationship between the trigonometric ratios of triangles xyz and abc. you must show all work and calculations to receive full credit. (5 points)
part b: explain how to find the measures of segments cb and ab. you must show all work and calculations to receive full credit. (5 points)

Step-by-step answer

17.02.2022, solved by verified expert

Nitin Arora

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Mathematics, English, Physics. Research Scholar at Indian Institute of Technology, Roorkee

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Answers:

Dilation is the process in which the dimensions of a given shape is increased or decreased by a scale factor.

Therefore the answers to the questions are:

A. Since the dilation of triangle XYZ do not affect the measure of its internal angles, then the trigonometric ratios of respective internal angles are the same.

B. segment CB = 10

segment AB = 11.18

Explanation:

From the given information in the question, triangle XYZ is an image of triangle ACB. In triangle XYZ, given that Sin X = ; it implies that its hypotenuse is 5.59, and the opposite side as 5. So that;

Sin X =

X = 0.8945

X =

Thus,

<X + <Y + <Z =

+ + <Z =

<Z = -

<Z =

So then, the third side can be determined by applying the Pythagoras theorem.

= +

= -

= 6.2481

x =

= 2.49962

x = 2.5

Thus,

Cos X =

Tan X =

Therefore:

A. Considering triangles XYZ and ACB, the measure of their respective internal angles are the same.

i.e <X ≅ <A

<B ≅ <Z

<C ≅ <Y

So that the trigonometric ratios of respective internal angles are the same.

Given that triangle XYZ was dilated by a scale factor of 2, then the respective length of each sides of triangle ACB is twice that of XYZ.

Then,

i. segment CB = 2 x segment YZ

= 2 x 5

segment CB = 10

ii. segment AB = 2 x segment XZ

= 2 x 5.59

segment AB = 11.18

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Mathematics

PLEASE HELP ! Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin X=5/5.59 Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ABC you must show all work and calculations to receive full credit. Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit.

Step-by-step answer

P Answered by Master

Part A: The corresponding trigonometry ratios of both triangles are equal.

Part B:

CB 10

AB = 5

Step-by-step explanation:

Original image = ∆XYZ

Dilated image = ∆ ACB

Scale factor = 2

Sin X=5/5.59

Find attached the diagram obtained from the given information.

Part A:

Sin X=opposite/hypotenuse = 5/5.59

YZ = 5, XY = 5.59

When a shape is dilated, the new shape has a similar shape but different size.

∆XYZ and ∆ ACB are similar triangles.

AC = scale factor × XY = 2XY

AB = scale factor × XZ = 2XZ

CB = scale factor × YZ = 2YZ

In similar triangles theorem, the ratio of their corresponding sides are equal.

The corresponding trigonometry ratios of both triangles are equal.

Sin X= YZ/XY

Cos X= XZ/XY

Tan X = YZ/XZ

In ∆ACB

Sin A= CB/AC = 2YZ/2XY

Cos A= AB/AC= 2XZ/2XY

Tan A = 2YZ/2XZ

SinA = SinX, CosA = CosX, TanA = TanX

Part B:

To find segment CB, we use the formula relating CB to the corresponding side YZ

CB = scale factor × YZ

CB = 2× 5 = 10

To find segment AB, we use the formula relating AB to the corresponding side XZ

AB = scale factor × XZ

AB = 2× XZ

To get XZ, we use Pythagoras theorem:

XY² = XZ² + YZ²

XZ² = 5.59² - 5²

XZ = √6.2481 = 2.5

AB = 2× 2.5 = 5

PLEASE HELP ! Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin X=5/5.5

Mathematics

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X = 5 over 5 and 59 hundredths. Triangles XYZ and ACB; angles Y and C both measure 90 degrees, angles A and X are congruent. Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ACB. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

Step-by-step answer

P Answered by Master

See explanation

Step-by-step explanation:

Given

$\triangle XYZ \simeq \triangle ACB$

$\sin X = \frac{5}{5.59}$

$\angle Y = \angle C = 90^o$

$\angle A \cong \angle X$

k = 2 --- scale factor

See attachment for triangles

Solving (a): The relationship between the angles of both triangles

The sine of an angle is:

$\sin(\theta) = \frac{Opposite}{Hypotenuse}$

By comparing the above to $\sin X = \frac{5}{5.59}$

$Opposite = 5\\ Hypotenuse = 5.59$

Using Pythagoras theorem:

Hypotenuse^2 = Opposite^2 + Adjacent^2

So, we have:

5.59^2 = 5^2 + Adjacent^2

31.2481 = 25 + Adjacent^2

Collect like terms

31.2481 - 25 =Adjacent^2

6.2481 =Adjacent^2

Take square roots

Adjacent = 2.50 --- approximated

For angle X, we have:

$Opposite = 5\\ Hypotenuse = 5.59\\ Adjacent = 2.50$

So, the relationship between both triangles are:

$\sin X =\sin A = \cos Z =\cos B= \frac{5}{5.59}$

$\cos X =\cos A =\sin Z =\sin B = \frac{2.50}{5.59}$

$\tan X =\tan A = \frac{5}{2.50} = 2$

$\tan Z =\tan B = \frac{2.50}{5} = \frac{1}{2}$

Solving (b): How to find CB and AB

In (a), we have:

For angle X, we have:

$Opposite = 5\\ Hypotenuse = 5.59\\ Adjacent = 2.50$

This implies that:

XY = 2.50

YZ = 5

XZ = 5.59

$\triangle XYZ$ is dilated by scale factor 2 to give $\triangle ACB$

So, CB and AB of $\triangle ACB$ are:

CB = 2 * YZ

CB = 2 * 5

CB = 10

AB = 2 * XZ

AB = 2 * 5.59

AB = 11.18

Mathematics

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X = 5 over 5 and 59 hundredths. Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ACB. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

Step-by-step answer

P Answered by Specialist

Bro, I attached your answer in attachment. Hope you got it. Thanks!!

Note: Pls click and see all the slides to get your clear answer.

Mathematics

80 POINT (ASAP) PLEASEEEE Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X = 5 over 5.59 Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ACB. You must show all work and calculations to receive full credit. Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

Step-by-step answer

P Answered by Master

Step-by-step explanation:

80 POINT (ASAP) PLEASEEEE

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB an

Mathematics

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X = 5 over 5 and 59 hundredths. Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ABC. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

Step-by-step answer

P Answered by Specialist

The angles of both triangles are the same. The sides are ratioed 1:2. y= 90 degrees

tan ∠X $=\frac{5}{2.5}$ . a^2+b^2=c^2.

Step-by-step explanation:

Mathematics

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X = 5 over 5 and 59 hundredths. Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ABC. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points) ALSO PLEASE GIVE AN ACTUAL ANSWER THIS HAS HAPPENED 3 TIMES FOR ME- Oop sorry caps but please actually

Step-by-step answer

P Answered by Specialist

The angles of both triangles are the same. The sides are ratioed 1:2. y= 90 degrees

tan ∠X $=\frac{5}{2.5}$ . a^2+b^2=c^2.

Step-by-step explanation:

Mathematics

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and sin ∠X = 5 over 5 and 59 hundredths. Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ABC. You must show all work and calculations to receive full credit. (5 points) Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

Step-by-step answer

P Answered by Specialist

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