D) What is the measure of ∠H? Explain. (1 point)

A certain 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 a

Scrat [10]

Answer:

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 $a$ 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

8 0

3 years ago

which expressions are equivalent to the first one? I don't understand how to determine that so please explain. Thanks!

coldgirl [10]

9514 1404 393

Answer:

(a) -(x+7)/y

(b) (x+7)/-y

Step-by-step explanation:

There are several ways you can show expressions are equivalent. Perhaps the easiest and best is to put them in the same form. For an expression such as this, I prefer the form of answer (a), where the minus sign is factored out and the numerator and denominator have positive coefficients.

The given expression with -1 factored out is ...

$\dfrac{-x-7}{y}=\dfrac{1(x+7)}{y}=\boxed{-\dfrac{x+7}{y}} \quad\text{matches A}$

Likewise, the expression of (b) with the minus sign factored out is ...

$\dfrac{x+7}{-y}=\boxed{-\dfrac{x+7}{y}}$

On the other hand, simplifying expression (c) gives something different.

$\dfrac{-x-7}{-y}=\dfrac{-(x+7)}{-(y)}=\dfrac{x+7}{y} \qquad\text{opposite the given expression}$

__

Another way you can write the expression is term-by-term with the terms in alpha-numeric sequence (so they're more easily compared).

Given: (-x-7)/y = (-x/y) +(-7/y)

(a) -(x+7)/y = (-x/y) +(-7/y)

(b) (x+7)/(-y) = (-x/y) +(-7/y)

(c) (-x-7)/(-y) = (x/y) +(7/y) . . . . not the same.

__

Of course, you need to know the use of the distributive property and the rules of signs.

a(b+c) = ab +ac

-a/b = a/(-b) = -(a/b)

-a/(-b) = a/b

__

Summary: The given expression matches (a) and (b).

_____

Additional comments

Sometimes, when I'm really stuck trying to see if two expressions are equal, I subtract one from the other. If the difference is zero, then I know they are the same. Looking at (b), we could compute ...

$\left(\dfrac{-x-7}{y}\right)-\left(\dfrac{x+7}{-y}\right)=\dfrac{-y(-x-7)-y(x+7)}{-y^2}\\\\=\dfrac{xy+7y-xy-7y}{-y^2}=\dfrac{0}{-y^2}=0$

Yet another way to check is to substitute numbers for the variables. It is a good idea to use (at least) one more set of numbers than there are variables, just to make sure you didn't accidentally find a solution where the expressions happen to be equal. We can use (x, y) = (1, 2), (2, 3), and (3, 5) for example.

The given expression evaluates to (-1-7)/2 = -4, (-2-7)/3 = -3, and (-3-7)/5 = -2.

(a) evaluates to -(1+7)/2 = -4, -(2+7)/3 = -3, -(3+7)/5 = -2, same as given

(b) evaluates to (1+7)/-2 = -4, (2+7)/-3 = -3, (3+7)/-5 = -2, same as given

(c) evaluates to (-1-7)/-2 = 4, different from given

3 0

3 years ago

(PLEASE HELP PLEASE!! IM GOING TO FAIL) PLEASE HELP ME WITH THESE QUESTIONS!! NOT JUST FOR ME FOR BRAIN! WHO WILL ANSWER FIRST?!

Lina20 [59]

1. A
2. should be 42?
3.???
4.27 apples left
5.B 1/4

7 0

3 years ago

(9\times100)+(8\times10)+\left (7\times\dfrac{1}{1000} \right )(9×100)+(8×10)+(7×

Tanya [424]

Answer:

7,980

Step-by-step explanation:

900 + 80 + (7 x 1000)

3 0

3 years ago

cody's Bikes rents bikes for $18 plus $5 per hour. Mark paid $48 to rent a bike. For how many hours did he rent the bike.

kkurt [141]

He rented the bike for 6 hours. If Cody charges an 18 dollar fee up front you subtract that from the total 48 to get 30. Then once you get down to the what he payed for the time used you divide it by 5 because that’s the hourly fee and get 6 hours

4 0

3 years ago

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