Given 3 sides of a triangle, compute interior angles and area

In summary: I like your solution better, but it wouldn't have shown him what was wrong. So, thanks for bringing that to my attention. :)The OP already has all three sides of the triangle, so Heron's Formula will be the most direct and exact way to get the area in this case...You actually make a good point here...it is preferable to use a formula that only relies on given data, not results we compute from this data. I wanted to show the OP that he was using an incorrect formula for the area. I like your solution better, but it wouldn't have shown him what was wrong. So, thanks for bringing that to my attention. :)Glad to have helped. I see what you mean about wanting to show
  • #1
ai93
54
0
Just attempted another exam question. Would you mind correcting me if I am wrong?
The question is
A triangle ABC has sides of length AB= 3.5 m, BC = 5.1 m and AC = 4.2m.

a) Calculate the size of the angle B and the size of the angle C, in degrees correct to 1 decimal place, in each case.

b) Calculate the area of triangle ABC.

My solution

\(\displaystyle a^{2}=b^{2}+c^{2}-2bcCOSA\)

\(\displaystyle \therefore COSA=\frac{b^{2}+c^{2}-a^{2}}{2bc}\)

\(\displaystyle COSA=\frac{4.2^{2}+3.5^{2}-5^{2}}{2\cdot4.2\cdot3.5}\)

A\(\displaystyle =COS^{-1}(0.132)\)
A= 82.4

Sine Rule

\(\displaystyle \frac{sin(82.4)}{5.1}=\frac{SinB}{4.2}\)

SinB=\(\displaystyle \frac{sin(82.4)\cdot4.2}{5.1}\)

B=Sin\(\displaystyle ^{-1}(0.816)

B=54.7\)

Now, 180-sin\(\displaystyle ^{-1}(0.816)=125.3\)
So angle B can equal 54.7 or 125.3

Angle C

180-(82.4+54.7)=42.9<180
or
180-(82.4+125.3)=-27.6 (which cannot be possible)
\(\displaystyle \therefore\)C = 42.9 degrees

and, 42.9+82.4+54.7=180

Question b)
Area=\(\displaystyle \frac{1}{2}bccosC\)

\(\displaystyle \frac{1}{2}\cdot5.1\cdot4.2 \cdot cos(42.9)\)=7.8m

Hope you understand that, I put what you told me into practice and hopefully solved this question correctly.
 
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  • #2
I have moved this post into a new thread...we ask that new questions not be tagged onto existing threads so that our threads don't become convoluted and hard to follow.

This is how I would work the problem:

First, we may use the Law of Cosines to find either angle B or C...I will find B first:

\(\displaystyle B=\arccos\left(\frac{3.5^2+5.1^2-4.2^2}{2\cdot3.5\cdot5.1}\right)=\arccos\left(\frac{1031}{1785}\right)\approx54.7^{\circ}\)

Next, use the Law of Sines to find angle C:

\(\displaystyle C=\arcsin\left(\frac{3.5\sin\left(\arccos\left(\dfrac{1031}{1785}\right)\right)}{4.2}\right)\approx42.9^{\circ}\)

There is no ambiguous case here, these are the only possible values.

Now, to find the area in square meters, we can use the formula:

\(\displaystyle A=\frac{1}{2}ab\sin(C)=\frac{1}{2}\cdot5.1\cdot4.2\cdot\frac{3.5\sin\left(\arccos\left(\dfrac{1031}{1785}\right)\right)}{4.2}=\frac{2}{25}\sqrt{8294}\approx7.3\)
 
  • #3
MarkFL said:
I have moved this post into a new thread...we ask that new questions not be tagged onto existing threads so that our threads don't become convoluted and hard to follow.

This is how I would work the problem:

First, we may use the Law of Cosines to find either angle B or C...I will find B first:

\(\displaystyle B=\arccos\left(\frac{3.5^2+5.1^2-4.2^2}{2\cdot3.5\cdot5.1}\right)=\arccos\left(\frac{1031}{1785}\right)\approx54.7^{\circ}\)

Next, use the Law of Sines to find angle C:

\(\displaystyle C=\arcsin\left(\frac{3.5\sin\left(\arccos\left(\dfrac{1031}{1785}\right)\right)}{4.2}\right)\approx42.9^{\circ}\)

There is no ambiguous case here, these are the only possible values.

Now, to find the area in square meters, we can use the formula:

\(\displaystyle A=\frac{1}{2}ab\sin(C)=\frac{1}{2}\cdot5.1\cdot4.2\cdot\frac{3.5\sin\left(\arccos\left(\dfrac{1031}{1785}\right)\right)}{4.2}=\frac{2}{25}\sqrt{8294}\approx7.3\)

The OP already has all three sides of the triangle, so Heron's Formula will be the most direct and exact way to get the area in this case...
 
  • #4
Prove It said:
The OP already has all three sides of the triangle, so Heron's Formula will be the most direct and exact way to get the area in this case...

Yep, that's another route one could take. :D
 
  • #5
Prove It said:
The OP already has all three sides of the triangle, so Heron's Formula will be the most direct and exact way to get the area in this case...

Gotten the angles correct, however got the area wrong.

I used this formula, \(\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)

Where s = \(\displaystyle \frac{a+b+c}{2}\)

\(\displaystyle \therefore s = \frac{5.1+4.2+3.5}{2}\)=6.4

\(\displaystyle \sqrt{6.4(6.4-5.1)(6.4-4.2)(6.4-3.5)}\)=7.28 \(\displaystyle \approx7.3m\)
 
  • #6
mathsheadache said:
Gotten the angles correct, however got the area wrong.

I used this formula, \(\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)

Where s = \(\displaystyle \frac{a+b+c}{2}\)

\(\displaystyle \therefore s = \frac{5.1+4.2+3.5}{2}\)=6.4

\(\displaystyle \sqrt{6.4(6.4-5.1)(6.4-4.2)(6.4-3.5)}\)=7.28 \(\displaystyle \approx7.3m\)
Why do you think it's wrong? It's pretty much the same as what Mark got...

Also, like I said, because you can keep the answer just as a surd (without any mention of angles), if you keep it in its surd form you will get the EXACT answer, not just a decimal...

$\displaystyle \begin{align*} \sqrt{6.4 \left( 6.4 - 5.1 \right) \left( 6.4 -4.2 \right) \left( 6.4-3.5 \right) } &= \sqrt{ 53.0816} \\ &= \sqrt{\frac{33\,176}{625}} \\ &= \frac{2\,\sqrt{8294}}{25} \end{align*}$

This is the same exact answer Mark got...
 
  • #7
Prove It said:
The OP already has all three sides of the triangle, so Heron's Formula will be the most direct and exact way to get the area in this case...

You actually make a good point here...it is preferable to use a formula that only relies on given data, not results we compute from this data. I wanted to show the OP that he was using an incorrect formula for the area.
 

FAQ: Given 3 sides of a triangle, compute interior angles and area

1. How do you find the interior angles of a triangle given three sides?

To find the interior angles of a triangle, you can use the Law of Cosines or the Law of Sines. The Law of Cosines states that for a triangle with sides a, b, and c, the cosine of one of the angles (let's call it angle A) can be found using the formula:
cos(A) = (b^2 + c^2 - a^2) / (2bc). You can then use inverse cosine (arccos) to find the value of angle A. Repeat this for the other two angles to find all three interior angles of the triangle.

2. What is the formula for finding the area of a triangle given three sides?

The formula for finding the area of a triangle given three sides is known as Heron's formula. It states that the area of a triangle (A) can be found using the formula:
A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter of the triangle and is found by adding all three sides and dividing by 2. This formula is useful when you only know the lengths of the sides of a triangle and not the height or base.

3. Can you find the interior angles and area of any triangle given three sides?

Yes, as long as the given lengths of the sides are valid (i.e. they follow the Triangle Inequality Theorem, where the sum of any two sides must be greater than the third side). However, there are certain cases where the given lengths may result in multiple solutions or no solution at all for the interior angles and/or area of the triangle.

4. Is it possible for a triangle to have three equal sides and still have different interior angles?

No, if a triangle has three equal sides (known as an equilateral triangle), then all three interior angles must also be equal. This is because the sum of all three angles in a triangle must be 180 degrees, and if all three sides are equal, all three angles must also be equal in order for the triangle to be closed.

5. Can you use the same formulas for finding the interior angles and area of a triangle with more than three sides?

No, the formulas for finding the interior angles and area of a triangle given three sides only apply to triangles with three sides. For polygons with more sides, you would need to use different formulas or methods specific to that shape.

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