Answer:
1) PM < PQ + QM
2) PM < PR + MR
3) PM < 1/2 (PQ + PR + QR)
Step-by-step explanation:
This problem is based on the Pythagorean Theorum Equation
which states that a right triangle with sides a, b and hypotenuse can be written mathematically as hypotenuse = [tex]\sqrt{a^2 + b^2}[/tex]
We can rewrite each line segment of the diagram into variables to simplify this problem and use the Pythagorean equation:
Line PQ = Hypotenuse 1, Line PR = Hypotenuse 2, Line PM = Side B, Line QM = Side A, Line MR = Side C
With all the line segments simplified into variables you can separate the diagram into 2 seperate triangles. Triangle 1 has H1, A, B and Triangle 2 has H2, C, B
Using the Pythagorean equation for both triangles you get:
H1 = [tex]\sqrt{a^2 + b^2}[/tex] and H2 = [tex]\sqrt {c^2 + b^2}[/tex]
Now you can assign random values for each variable and compare the problem statements.
For Triangle 1 lets assign variables A = 1, B = 2, and H1 = [tex]\sqrt{1^2 + 2^2}[/tex] = [tex]\sqrt{3}[/tex] = 1.73 For Triangle 2 lets assign variables C = 3, B = 2 and H2 = [tex]\sqrt {3^2 + 2^2} = \sqrt {13}[/tex] = 3.6
For the final step you can directly compare the statements and see which symbol is correct.
1) B ____ H1 + A is equivalent to 2 ____ (1.73 + 1), which is turns out to be 2 < 2.73 The answer is <
2) B ____ H2 + C is equivalent to 2 ____ (3.6 + 3), which turns out to be
2 < 6.6 The answer is < again
3) B ____ 1/2 (H1 + H2 + (A+C) is equivalent to 2 _____ [1/2] (1.73 + 3.6 + 1 + 3), which turns out to be 2 < 4.67 The answer is < again
