Calculate the area of a triangle knowing its perimeter and 2 heights

In summary, the conversation discusses the process of calculating the area of a triangle given its perimeter and two heights. The equation used involves equalizing two different formulas for the area and solving for one unknown. The final answer is option B.
  • #1
loquetedigo
14
0
Calculate the area of a triangle knowing its perimeter and 2 heights

perimeter = 30 m
ha = 8 m
hb = 9 mNOTE = You can use the online triangle calculator TrianCal to see and draw the results.
NOTE = Do not use the values ??of responses.

A) 41.29 m2
B) 42.93 m2 or 36.28 m2
C) 42.95 m2 or 36.29 m2
D) Imposible
 
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  • #2
Hello, here is what I did (this was the first idea that came to mind and it's kind of a brute force method):

Assuming the sides of the triangle are a,b,c, we have the following equations:

a+b+c+=30

The area of the triangle can be written in 3 different ways.

2 of them using A = Base * Height / 2

A = 8a/2

A = 9b/2

3rd one using Heron's Formula:

Semi-perimeter is 30/2 = 15 so

A = \sqrt{15(15-a)(15-b)(15-c)}

Now, to get an equation in only one unknown, let's pick a.

From equalizing the first 2 formulas for the area, we should get

b = 8a/9

From the perimeter, substituting b and solving for c we should get

c=(270-17a)/9

You should be able to get these answers fairly easy yourself.

Lastly, to write the equation, we equalize the Area that contains a (1st one) and the one from Heron's formula, where we substitute b and c with the values found previously. After simplifying, we should get something like this:

36a=\sqrt{15(15-a)(135-8a)(17a-135)}.

I did't actually bother with trying to solve the equation, but I did graph it using Desmos and found 3 solutions for a
9.071
10.732
20.649

Ignoring the last one (because c would be negative) and substituting both values in the Area formula, after rounding the final answer should be B.
 
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FAQ: Calculate the area of a triangle knowing its perimeter and 2 heights

1. How do you calculate the area of a triangle given its perimeter and 2 heights?

To calculate the area of a triangle, you can use the formula A = (b*h)/2, where b is the base of the triangle and h is the corresponding height. In this case, you have 2 heights, so you will need to add them together and use the corresponding side length as the base. Once you have the base and height, plug them into the formula to calculate the area.

2. What is the formula for finding the base length of a triangle given its perimeter and 2 heights?

The formula for finding the base length of a triangle is b = (2A)/h, where A is the area of the triangle and h is the corresponding height. In this case, you will need to use the calculated area and add the two heights together to find the base length.

3. Can you provide an example of how to calculate the area of a triangle using this method?

Yes, for example, if a triangle has a perimeter of 24 units and two heights of 4 and 7 units, we can calculate the area as follows:
Step 1: Add the two heights together (4+7=11)
Step 2: Subtract this sum from the perimeter (24-11=13)
Step 3: Divide this result by 2 to get the base length (13/2=6.5)
Step 4: Plug in the base (6.5) and one of the heights (4) into the formula A = (b*h)/2
Step 5: Calculate the area (A = (6.5*4)/2 = 13 square units)
Therefore, the area of the triangle is 13 square units.

4. Can this method be used for all types of triangles?

Yes, this method can be used for all types of triangles, as long as you have the perimeter and at least two heights to calculate the area.

5. Are there any other methods for calculating the area of a triangle?

Yes, there are other methods for calculating the area of a triangle, such as using the lengths of all three sides (Heron's formula) or using trigonometric functions. However, the method of using the perimeter and two heights is a simple and efficient way to calculate the area of a triangle.

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