Which value from the list below, when substituted for x, would represent an acute triangle with side lengths x, x+ 4, and 20? Assume that the longest side of the triangle is of length 20 units.
8
10
12
14

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. (Think about it, if one side was the same length as the other two combined that would make a straight line. If it was any longer the other two sides couldn't reach the endpoints.)

This means:

[tex]x+(x+4)>20\\2x+4>20\\2x>16\\x>8[/tex]

Now you also said the triangle is acute. That means that all angles are less than 90°. Since 20 is the longest side length, the angle across from the side with length 20 is the biggest angle. So x has to be less than whatever values make that angle exactly 90. So just find the values of x when the angle opposite the side with length 20 is 90.

The Pythagorean theorem says:

[tex]x^{2} +(x+4)^{2} =20^{2} \\x^{2} +x^{2} +8x+16=400\\2x^{2} +8x-384=0\\x^{2} +4x-192=0\\x^{2} +16x-12x-192=0\\x(x+16)-12(x+16)=0\\x+16 = 0, x-12=0\\x=-16, 12[/tex]

So x has to be less than 12. x < 12

Therefore, [tex]8<x<12[/tex]

For more information:

https://brainly.com/question/16664220

Which value from the list below, when substituted for x, would represent an acute triangle with side lengths x, x+ 4, and 20? Assume that the longest side of the triangle is of length 20 units. 8 10 12 14

Respuesta :

Otras preguntas

Which value from the list below, when substituted for x, would represent an acute triangle with side lengths x, x+ 4, and 20? Assume that the longest side of the triangle is of length 20 units.
8
10
12
14