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Elena-2011 [213]
3 years ago
11

The image below is made from a regular octagon and a regular triangle. What is the value of x?

Mathematics
1 answer:
notsponge [240]3 years ago
7 0
For some reason I think it’s 60 but hopefully I’m right?! シ
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Work out the area of abcd.<br><br> please ensure you give workings out too.
ipn [44]

Answer:

\displaystyle A_{\text{Total}}\approx45.0861\approx45.1

Step-by-step explanation:

We can use the trigonometric formula for the area of a triangle:

\displaystyle A=\frac{1}{2}ab\sin(C)

Where a and b are the side lengths, and C is the angle <em>between</em> the two side lengths.

As demonstrated by the line, ABCD is the sum of the areas of two triangles: a right triangle ABD and a scalene triangle CDB.

We will determine the area of each triangle individually and then sum their values.

Right Triangle ABD:

We can use the above area formula if we know the angle between two sides.

Looking at our triangle, we know that ∠ADB is 55 DB is 10.

So, if we can find AD, we can apply the formula.

Notice that AD is the adjacent side to ∠ADB. Also, DB is the hypotenuse.

Since this is a right triangle, we can utilize the trig ratios.

In this case, we will use cosine. Remember that cosine is the ratio of the adjacent side to the hypotenuse.

Therefore:

\displaystyle \cos(55)=\frac{AD}{10}

Solve for AD:

AD=10\cos(55)

Now, we can use the formula. We have:

\displaystyle A=\frac{1}{2}ab\sin(C)

Substituting AD for a, 10 for b, and 55 for C, we get:

\displaystyle A=\frac{1}{2}(10\cos(55))(10)\sin(55)

Simplify. Therefore, the area of the right triangle is:

A=50\cos(55)\sin(55)

We will not evaluate this, as we do not want inaccuracies in our final answer.

Scalene Triangle CDB:

We will use the same tactic as above.

We see that if we can determine CD, we can use our area formula.

First, we can determine ∠C. Since the interior angles sum to 180 in a triangle, this means that:

\begin{aligned}m \angle C+44+38&=180 \\m\angle C+82&=180 \\ m\angle C&=98\end{aligned}

Notice that we know the angle opposite to CD.

And, ∠C is opposite to BD, which measures 10.

Therefore, we can use the Law of Sines to determine CD:

\displaystyle \frac{\sin(A)}{a}=\frac{\sin(B)}{b}

Where A and B are the angles opposite to its respective sides.

So, we can substitute 98 for A, 10 for a, 38 for B, and CD for b. Therefore:

\displaystyle \frac{\sin(98)}{10}=\frac{\sin(38)}{CD}

Solve for CD. Cross-multiply:

CD\sin(98)=10\sin(38)

Divide both sides by sin(98). Hence:

\displaystyle CD=\frac{10\sin(38)}{\sin(98)}

Therefore, we can now use our area formula:

\displaystyle A=\frac{1}{2}ab\sin(C)

We will substitute 10 for a, CD for b, and 44 for C. Hence:

\displaystyle A=\frac{1}{2}(10)(\frac{10\sin(38)}{\sin(98)})\sin(44)

Simplify. So, the area of the scalene triangle is:

\displaystyle A=\frac{50\sin(38)\sin(44)}{\sin(98)}

Therefore, our total area will be given by:

\displaystyle A_{\text{Total}}=50\cos(55)\sin(55)+\frac{50\sin(38)\sin(44)}{\sin(98)}

Approximate. Use a calculator. Thus:

\displaystyle A_{\text{Total}}\approx45.0861\approx45.1

8 0
3 years ago
Rita has already prepared 3 kilograms of dough and will continue preparing 1 kilogram of dough every hour. How much dough did Ri
Digiron [165]

Answer:

12

Step-by-step explanation:

9+3=12

hope it helps

btw I need one more brainliest to level up

6 0
3 years ago
Please help me because I need to turn this on tommorrow
Trava [24]
YAY I KNOW HOW TO DO THIS OKAY
so 90 degrees minus 49 degrees equals 41 degrees
then 41= (x+3) then you subtract 3
so x=38 
7 0
3 years ago
Read 2 more answers
Solve each question for X , y=-9+x^2
aleksandrvk [35]
So basically you want to isolate only one of the variable on one side so
y=-9+x^2
add 9 to both sides
y+9=x^2
square root both sides
\sqrt{y+9}=x

x=\sqrt{y+9}
5 0
3 years ago
Read 2 more answers
Hello need help please with Divide rational and simplify
aliina [53]
Dividing fractions is the same as multiplying by its reciprocal so you can change the equation to this
\frac{ - 14 + 7x}{4}  \times  \frac{4}{4 - 2x}
And you can simplify by eliminating the fours and multiplying straight across
\frac{ - 14 + 7x}{4 - 2x}
You can pull out - 7 from the top and 2 from the bottom so you have this
\frac{ - 7(2  -  x)}{2(2 - x)}
And then you can cancel out to (2-x) to get an answer of
\frac{ - 7}{2}
5 0
2 years ago
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