Question

Work out the area of abcd.

please ensure you give workings out too.

Answer 1

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

Explanation:

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

`\displaystyle A=(1)/(2)ab\sin(C)`

Where a and b are the side lengths, and C is the angle between 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)=(AD)/(10)`

Solve for AD:

`AD=10\cos(55)`

Now, we can use the formula. We have:

`\displaystyle A=(1)/(2)ab\sin(C)`

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

`\displaystyle A=(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 (\sin(A))/(a)=(\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 (\sin(98))/(10)=(\sin(38))/(CD)`

Solve for CD. Cross-multiply:

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

Divide both sides by sin(98). Hence:

`\displaystyle CD=(10\sin(38))/(\sin(98))`

Therefore, we can now use our area formula:

`\displaystyle A=(1)/(2)ab\sin(C)`

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

`\displaystyle A=(1)/(2)(10)((10\sin(38))/(\sin(98)))\sin(44)`

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

`\displaystyle A=(50\sin(38)\sin(44))/(\sin(98))`

Therefore, our total area will be given by:

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

Approximate. Use a calculator. Thus:

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