Luke takes a sheet of paper and cuts from one corner to the opposite corner, making two triangles. if the piece of paper is 16 inches long and 12 inches wide, how long is the dagonal cut that luke made? make sure to show your work and explain ypur answer.

You need to apply the Pythagoras theorem, which is a squared + b squared is c squared. C squared in this formula is the hypotenuse squared. The hypotenuse is the longest side of a triangle, opposite the largest angle of the triangle. Keep in mind that the Pythagoras theorem only applies to right angle triangles. 

In this question c squared is the diagonal you are trying to figure out.

A squared is any side of the triangle other than c squared, and b squared is the other side of the triangle. So a squared is 16 inches squared, and b squared is 12 inches squared.

To work this out, we square 16 to give 256.

Then 12 is squared - 144.

Then we square root 256 + 144, or the square root of 400.

The square root of 400 is 20, so the diagonal Luke cut was 20 inches long.

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and y≤ -x +4 .

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

.

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

y≤ -x +4 .