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2 years ago

If triangles ABC and XYZ are similar, what is the length of side b?

Mathematics

1 answer:

Semmy [17]2 years ago

7 0

Length of side b = 12

By doing 15 / 10, we can find how much the right triangle has been reduced compared to the left triangle. So, it’s been reduced by 0.5

Because we’re trying to find side b which is on the larger (left) triangle, we do 8 x 1.5 which is 12

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let sin(θ) =3/5 and tan(y) =12/5 both angels comes from 2 different right trianglesa)find the third side of the two tringles b)

statuscvo [17]

In a right triangle, we haev some trigonometric relationships between the sides and angles. Given an angle, the ratio between the opposite side to the angle by the hypotenuse is the sine of this angle, therefore, the following statement

$\sin (\theta)=\frac{3}{5}$

Describes the following triangle

To find the missing length x, we could use the Pythagorean Theorem. The sum of the squares of the legs is equal to the square of the hypotenuse. From this, we have the following equation

$x^2+3^2=5^2$

Solving for x, we have

$\begin{gathered} x^2+3^2=5^2 \\ x^2+9=25 \\ x^2=25-9 \\ x^2=16 \\ x=\sqrt[]{16} \\ x=4 \end{gathered}$

The missing length of the first triangle is equal to 4.

For the other triangle, instead of a sine we have a tangent relation. Given an angle in a right triangle, its tanget is equal to the ratio between the opposite side and adjacent side.The following expression

$\tan (y)=\frac{12}{5}$

Describes the following triangle

Using the Pythagorean Theorem again, we have

$5^2+12^2=h^2$

Solving for h, we have

$\begin{gathered} 5^2+12^2=h^2 \\ 25+144=h^2 \\ 169=h^2 \\ h=\sqrt[]{169} \\ h=13 \end{gathered}$

The missing side measure is equal to 13.

Now that we have all sides of both triangles, we can construct any trigonometric relation for those angles.

The sine is the ratio between the opposite side and the hypotenuse, and the cosine is the ratio between the adjacent side and the hypotenuse, therefore, we have the following relations for our angles

$\begin{gathered} \sin (\theta)=\frac{3}{5} \\ \cos (\theta)=\frac{4}{5} \\ \sin (y)=\frac{12}{13} \\ \cos (y)=\frac{5}{13} \end{gathered}$

To calculate the sine and cosine of the sum

$\begin{gathered} \sin (\theta+y) \\ \cos (\theta+y) \end{gathered}$

We can use the following identities

$\begin{gathered} \sin (A+B)=\sin A\cos B+\cos A\sin B \\ \cos (A+B)=\cos A\cos B-\sin A\sin B \end{gathered}$

Using those identities in our problem, we're going to have

$\begin{gathered} \sin (\theta+y)=\sin \theta\cos y+\cos \theta\sin y=\frac{3}{5}\cdot\frac{5}{13}+\frac{4}{5}\cdot\frac{12}{13}=\frac{63}{65} \\ \cos (\theta+y)=\cos \theta\cos y-\sin \theta\sin y=\frac{4}{5}\cdot\frac{5}{13}-\frac{3}{5}\cdot\frac{12}{13}=-\frac{16}{65} \end{gathered}$

4 0

1 year ago

Express the following numbers in standard form. [A] 509.78 [B] 0.0289 Can someone please help me ? I need to answer that questio

STALIN [3.7K]

Answer:

(A) 509.78

We need to move the decimal point after the first non-zero digit

So,

=> 5.0978 × 10² (Since the decimal point is moved two places to the left)

(B) 0.0289

Again after the first non-zero digit:

So, it becomes:

=> 2.89 × 10⁻² (Since the decimal point is moved 2 places to the right so it will be negative also)

8 0

3 years ago

Read 2 more answers

The interior angles of a triangle have measures k°, 27°, and 10°.<br><br> What is the value of k?

NikAS [45]

Hey!

Hope this helps...

~~~~~~~~~~~~~~~~~~~~~

Assuming that the sum of all the angles must equal 180°

180 - (27+10) = k

180 - 37 = k

k = 143°

6 0

3 years ago

Transform the equation to isolate x: ax=bx+1. How is the value of x related to the difference of a and b

nasty-shy [4]

Answer:

x is the reciprocal of the difference of a and b.

Step-by-step explanation:

ax=bx+1

Subtract by bx on both sides

ax-bx=1

x(a-b)=1

x= 1/(a-b)

8 0

2 years ago

Read 2 more answers

A granite block in the shape of a right rectangular

natali 33 [55]

Here is the set up:

Let m = mass of the block

m = (30•40•50)/(2.8)

Use your calculator to calculate the left side for m.

6 0

3 years ago