an isosceles triangle has two sides of length p and one of length m.

Question

In terms of these lengths, write calculator-ready formulas for the sizes of the angles of this triangle.

 

 

 

 

Can someone please help me with this problem, please.

Answer 1

Hi Moxa.

 

If you have two side-lengths of p and one side-length of m, you can draw a vertical line from the top of the triangle to the base. This will create two equal right triangles with hypotenuse p and base 1/2*m.

 

The base angles can be determined by:

theta_base = arcos(m/2p)

 

The top angle can be determined by:

 

theta_top = 2*arcsin(m/2p)

 

You multiply by two to find theta_top because by drawing the vertical you cut the angle of your original isosceles triangle in half. So arcsin(m/2p) would only find half of the original angle measure.

 

Hope this helps.

Answer 2

Hi Moxa,

 

Lets construct this isosceles triangle such that the side of length m is along the horizontal and the other two sides of length p sprout upward from the horizontal. I have tried to draw a picture of this triangle. Let's consider angles A, B and C as shown. 

 

 

      /\

 p  /C \ p

   /      \

  /A   B \

  -------

      m

 

 

We can use the law of cosines to determine these angles. The next triangle is a general triangle:

 

      /\
 b  /C \ a
   /      \
  /A   B \
  -------
      c

 

You can apply the Law of Cosines three different ways:

 

a² = b² + c² - 2bcos(A)

 

b² = a² + c² -2accos(B)

 

c² = a² + b² -2bccos(C)

 

For our particular specific triangle, we use the Law of Cosines in the following manner:

 

p² = p² + m² - 2pmcos(A)

 

The p²'s cancel out. So we have

 

2pmcos(A) = m²

 

We can cancel out an m. And we get

 

2pcos(A) = m

 

or

 

A = arcos(m/(2p))

 

Calculating the angle B is the exact same Math. So, we have

 

B = arcos(m/(2p))

 

Finally, we need to calculate angle C. Using the Law of Cosines:

 

m² = p² + p² -2p*pcos(C)

 

or

 

m² = 2p² -2p²cos(C)

 

or

 

2p²cos(C) = 2p² - m²

 

C = arcos[(2p² - m²)/(2p²)]

 

So, altogether, your calculator-ready formulas are:

 

A = arcos(m/(2p))

 

B = arcos(m/(2p))

 

C = arcos[(2p² - m²)/(2p²)]