The length of a rectangle plus its width is 15 cm. The area is 44 square cm. What are the length and width of the rectangle?
2 answers:
This problem requires a system of equations, which we can easily come up with. "The length plus the width is 15" is L + W = 15. That is the first of 2 equations. The area is found by multiplying the length by the width, so the next statement, "The area is 44" is L*W=44. Solve that first equation for L to get L=15-W. Now sub that L value into the second equation in place of L to get an equation in terms of W only. (15-W)W=44. We will distribute to get 
. Now we will bring the 44 ovr by subtraction and put this into standard form: 
. I personally do not like to try to factor polynomials with negative leading coefficients, so let's change all those signs (yes, that's legal): 
. If you plug those values into the quadratic formula you will get w values of 11 and 4. It just so happens (and NOT by coincidence) that 11+4 = 15, which fits our "the length of a rectangle plus its width is 15", and 11*4 = 44, which fits our "the area of the rectangle is 44". So the length and width are 11 and 4, respectively.
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Since you're only asked the ordered pair of D'', it's much easier just to plot and reflect point D twice than to do that for all four points!
Remember that reflecting points is like putting a mirror at the line of reflection or flipping that point over at that line. The reflected point should be the same distance from the line of reflection as the original point.
1) Reflect D over the x-axis to get D'.
D is at (4,1). Draw a line that is perpendicular to the line of reflection and goes through D. D is as far from the line of reflection as D' should be on its other side (both are on that perpendicular line). Since D is 1 unit above the x-axis, that means D' is 1 unit below at (4, -1). See picture 1.
2) Reflect D' over <span>y=x+1 to get D''.
D' is at (4, -1). Draw </span>y=x+1 and the line perpendicular to it going through D''. D'' is the same distance from the line of reflection on the other side. See picture 2. D'' is at (-2, 5).
Answer: D'' is at (-2, 5)
Remember that reflecting points is like putting a mirror at the line of reflection or flipping that point over at that line. The reflected point should be the same distance from the line of reflection as the original point.
1) Reflect D over the x-axis to get D'.
D is at (4,1). Draw a line that is perpendicular to the line of reflection and goes through D. D is as far from the line of reflection as D' should be on its other side (both are on that perpendicular line). Since D is 1 unit above the x-axis, that means D' is 1 unit below at (4, -1). See picture 1.
2) Reflect D' over <span>y=x+1 to get D''.
D' is at (4, -1). Draw </span>y=x+1 and the line perpendicular to it going through D''. D'' is the same distance from the line of reflection on the other side. See picture 2. D'' is at (-2, 5).
Answer: D'' is at (-2, 5)