Mathematics, 12.09.2019 20:20 tonyiacamper1

Acertain square is to be drawn on a coordinate plane. one of the vertices must be on the origin, and the square is to have an area of 100. if all coordinates of the vertices must be integers, how many different ways can this square be drawn?
(a) 4
(b) 6
(c) 8
(d) 10
(e) 12

Answers: 1

Show answers

Answers

Answer from: ricardodeleon152

The answer is (C) 8

Step-by-step explanation:

First, let's calculate the length of the side of the square.

$A_{square}=a^2$ , where is the length of the side. Now, let's try to build the square. First we need to find a point which distance from (0, 0) is 10. For this, we can use the distance formula in the plane:

$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ which for x_1=0 and y_1 = 0 transforms as $d=\sqrt{(x_2)^2 + (y_2)^2}$ . The first point we are looking for is connected to the origin and therefore, its components will form a right triangle in which, the Pythagoras theorem holds, see the first attached figure. Then, x_2 , y_2 and 10 are a Pythagorean triple. From this, x_2= 6 or x_2=8 while y_2= 6 or y_2=8 . This leads us with the set of coordinates:

$(\pm 6, \pm 8)$ and $(\pm 8, \pm 6)$ . (A)

The next step is to find the coordinates of points that lie on lines which are perpendicular to the lines that joins the origin of the coordinate system with the set of points given in (A):

Let's do this for the point (6, 8).

The equation of the line that join the point (6, 8) with the origin (0, 0) has the equation y = mx +n , however, we only need to find its slope in order to find a perpendicular line to it. Thus,

$m = \frac{y_2-y_1}{x_2-x_1} \\m = \frac{8-0}{6-0} \\m = 8/6$

Then, a perpendicular line has an slope $m_{\bot} = -\frac{1}{m} = -\frac{6}{8}$ (perpendicularity condition of two lines). With the equation of the slope of the perpendicular line and the given point (6, 8), together with the equation of the distance we can form a system of equations to find the coordinates of two points that lie on this perpendicular line.

$m_{\bot}=\frac{6}{8} = \frac{8-y}{6-x}\\ 6(6-x)+8(8-y)=0$ (1)

$d^2 = \sqrt{(y_o-y)^2+(x_o-x)^2} \\(10)^2=\sqrt{(8-y)^2+(6-x)^2}\\100 = \sqrt{(8-y)^2+(6-x)^2}$ (2)

This system has solutions in the coordinates (-2, 14) and (14, 2). Until here, we have three vertices of the square. Let's now find the fourth one in the same way we found the third one using the point (14,2). A line perpendicular to the line that joins the point (6, 8) and (14, 2) has an slope m = 8/6 based on the perpendicularity condition. Thus, we can form the system:

$\frac{8}{6} =\frac{2-y}{14-x} \\8(14-x) - 6(2-y) = 0$ (1)

$100 = \sqrt{(14-x)^2+(2-y)^2}$ (2)

with solution the coordinates (8, -6) and (20, 10). If you draw a line joining the coordinates (0, 0), (6, 8), (14, 2) and (8, -6) you will get one of the squares that fulfill the conditions of the problem. By repeating this process with the coordinates in (A), the following squares are found:

(0, 0), (6, 8), (14, 2), (8, -6)(0, 0), (8, 6), (14, -2), (6, -8)(0, 0), (-6, 8), (-14, 2), (-8, -6)(0, 0), (-8, 6), (-14, -2), (-6, -8)

Now, notice that the equation of distance between the two points separated a distance of 10 has the trivial solution $(\pm10, 0)$ and $(0, \pm10)$ . By combining this points we get the following squares:

(0, 0), (10, 0), (10, 10), (0, 10)(0, 0), (0, 10), (-10, 10), (-10, 0)(0, 0), (-10, 0), (-10, -10), (0, -10)(0, 0), (0, -10), (-10, -10), (10, 0)

See the attached second attached figure. Therefore, 8 squares can be drawn

Acertain square is to be drawn on a coordinate plane. one of the vertices must be on the origin, and

Answer from: Quest

Iam thinking the answer is a.) knew

Answer from: Quest

then say something, tell them they won’t respect you if you don’t even have respect for your self sweetheart you need to lay it on them, all of it : )

Another question on Mathematics

Mathematics, 21.06.2019 20:10

Ascientist has discovered an organism that produces five offspring exactly one hour after its own birth, and then goes on to live for one week without producing any additional offspring. each replicated organism also replicates at the same rate. at hour one, there is one organism. at hour two, there are five more organisms. how many total organisms are there at hour seven? 2,801 19,531 19,607 97.655

Answers: 1

Answer

Mathematics, 21.06.2019 20:30

Venita is sewing new curtains. the table below shows the relationship between the number of stitches in a row and the length ofthe row.sewingnumber of stitches length (cm)51 131532024306what is the relationship between the number of stitches and the length? othe number of stitches istimes the length.o. the number of stitches is 3 times the length.

Answers: 2

Answer

Mathematics, 21.06.2019 22:00

Billie holiday sells children's clothing for sara vaughn's toddler shop. baby blankets sell for $29.99 after a markup rate based on cost of 109%. find the cost and the markup.

Answers: 2

Answer

Mathematics, 21.06.2019 22:00

The coordinates r(1, -3), s(3, -1) t(5, -7) form what type of polygon? a right triangle an acute triangle an equilateral triangle an obtuse triangle

Answers: 1

Answer

You know the right answer?

Acertain square is to be drawn on a coordinate plane. one of the vertices must be on the origin, and...

Questions