Area of triangle inside parallelogram

[cjwalle]

Homework Statement

http://folk.uio.no/robinbj/gg/ggstart.pdf"
I am supposed to find the area of the triangle PQD. The numbers given are the areas of the other triangles.

Homework Equations

$$A= \frac{1}{2} a b \sin{C}$$

As well as Heron's formula, possibly?
$$A= \sqrt{s(s-a)(s-b)(s-c)}$$ where $$s = \frac{a+b+c}{2}$$

The Attempt at a Solution

Where I'm stumped is exactly how to start. At first, I figured I'd try to find the values of a, b and c. Using the area of a triangle:

$$a = DP = \frac{14}{AP\sin{\theta}}$$

Similarly, for b:

$$b = QP = \frac{56}{QB\sin{\alpha}}$$

And c:

$$c = QD = \frac{28}{QC\sin{x}}$$

Meaning that $$s = \frac{7}{AP\sin{\theta}} + \frac{28}{QB\sin{\alpha}} + \frac{14}{QC\sin{x}}$$

However, dealing with three different angles, without knowing the relationship between them or the sum of the angles, I just don't know how to proceed and solve the exercise based on this. I am not looking for a solution from you guys, mind you. Just a tip to get me on the right track?

Thank you.

 

[tiny-tim]

Hi cjwalle!

I don't think the angles matter …

won't the ratios be the same, whatever the angles are, so you might as well make it a square?

 

[Architeuthis Dux]

my first step would be to carefully analyze the given information and identify any patterns or relationships that may be useful in solving the problem. In this case, we can see that the areas of the triangles PQA and PQC are given and we know that the area of a triangle is half the product of its base and height. This means that we can express the height of these triangles in terms of their bases and areas.

Next, I would draw a diagram and label all known and unknown quantities, using the relationships we have identified. This will help us visualize the problem and make it easier to identify a possible solution.

Once we have a clear understanding of the given information and have a diagram to guide us, we can start considering different mathematical methods to solve the problem. As you have mentioned, we can use the formula for the area of a triangle (A= \frac{1}{2} a b \sin{C}) or Heron's formula (A= \sqrt{s(s-a)(s-b)(s-c)} where s = \frac{a+b+c}{2}).

In this case, it may be helpful to use Heron's formula, as it allows us to calculate the area of a triangle using only the lengths of its sides. We can use the values we have calculated for a, b, and c to determine the semi-perimeter (s) and then plug those values into the formula to find the area of triangle PQD.

In addition, we can also use the fact that the area of a parallelogram is equal to the product of its base and height. This means that we can find the area of parallelogram PQCB using the given information and then subtract the areas of triangles PQA and PQC to find the area of triangle PQD.

Overall, it is important to carefully analyze the given information, draw a diagram, and consider different mathematical methods to find the solution. Good luck with your homework!