Question

Using Pythagorean inequalities, determine which of the following sets of values can represent the lengths of the sides of an acute triangle.
a. 1,2,3
b. 6,8,10
c. 6,7,8
d. 4,6,10

Answer 1

Answer:

c. 6, 7, 8

Step-by-step explanation:

$a,\ b,\ c-\text{sides of a triangle}\\\\\text{If}\ a\leq b< c\ \text{and}\ a^2+b^2=c^2,\ \text{then it's the lengths of the sides of a right triangle}\\\\\text{If}\ a\leq b\leq c\ \text{and}\ a^2+b^2>c^2,\ \text{then it's the lengths of the sides of a acute triangle}\\\\\text{If}\ a\leq b< c\ \text{and}\ a^2+b^2$

$\text{Triangle condition:}\\\text{That each side has to be shorter than the sum of the other two sides}\\\text{and longer than their difference.}\\\\\text{If}\ a\leq b\leq c,\ \text{then}\ a+b>c.$

$--------------------------\\a.\\a=1,\ b=2,\ c=3\\\\a+b=1+2=3=c\\\\\text{These are not the lengths of the sides of the triangle.}\\----------------------------\\b.\\a=6,\ b=8,\ c=10\\\\a+b=6+8=14>10=c\qquad OK\\\\a^2+b^2=6^2+8^2=36+64=100\\\\c^2=10^2=100\\\\a^2+b^2=c^2\\\\\text{It's the right triangle.}\\----------------------------$

$c.\\a=6,\ b=7,\ c=8\\\\a+b=6+7=13>8\qquad OK\\\\a^2+b^2=6^2+7^2=36+49=85\\\\c^2=8^2=64\\\\a^2+b^2>c^2\\\\\text{It's the acute triangle.}\\----------------------------\\d.\\a=4,\ b=6,\ c=10\\\\a+b=4+6=10=c\\\\\text{These are not the lengths of the sides of the triangle.}$