Mathematics, 10.03.2020 18:48 owen74

In the coordinate plane the vertices of rst are: r(6 -1), s (1 -4), t (-5,6). Prove that rst is a right triangle. State the coordinates of p such that quadrilateral rstp is a rectangle.

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Answer from: efjdjskd

The triangle rst is a right triangle, and the coordinate of p such that rstp is a rectangle is (-10,3)

The coordinates of the triangle are given as:

r = (6 -1)

s= (1 -4)

t= (-5,6)

Start by calculating the slopes of the sides of the triangle using:

$m = \frac{y_2 -y_1}{x_2 -x_1}$

So, we have:

$rs = \frac{-4--1}{1-6}$

$rs = \frac{3}{5}$

$st = \frac{6--4}{-5-1}$

$st = \frac{10}{-6}$

Simplify

$st = -\frac{5}{3}$

For the triangle to be a right triangle, then the product of the slopes must equal -1.

This is represented as:

$m_1 \times m_2 = -1$

So, we have:

$rs \times st = -1$

Substitute values for rs and st

$\frac 35 \times -\frac 53 = -1$

-1 =-1

The above equation is true.

Hence, triangle rst is a right triangle

r = (6 -1)

s= (1 -4)

t= (-5,6)

Calculate the distance rs using:

$rs = \sqrt{(x_1 -x_2)^2 + (y_1 -y_2)^2}$

$rs = \sqrt{(6 -1)^2 + (-1--4)^2}$

Calculate the distance tp using:

$tp = \sqrt{(x_1 -x_2)^2 + (y_1 -y_2)^2}$

So, we have:

$tp = \sqrt{(-5 -x)^2 + (6-y)^2}$

Opposite sides are equal.

So, we have:

$\sqrt{(-5 -x)^2 + (6-y)^2} = \sqrt{(6 -1)^2 + (-1--4)^2}$

Take square of both sides

(-5 -x)^2 + (6-y)^2 = (6 -1)^2 + (-1--4)^2

By comparison, we have:

(-5 -x)^2 = (6 -1)^2

and

(6-y)^2 = (-1--4)^2

For (-5 -x)^2 = (6 -1)^2

Take square root of both sides

-5 -x = 6 - 1

Add 5 to both sides

-x = 6 - 1 + 5

-x = 10

So, we have:

x = -10

Also, we have:

(6-y)^2 = (-1--4)^2

Take square root of both sides

6 - y = -1--4

6 - y = 3

Collect like terms

y = 6-3

y =3

So, we have:

x = -10 and y =3

Hence, the coordinate of p is (-10,3)

Read more about coordinates at:

link

Answer from: madisonanders

Coordinates of point p: (0,9)

Step-by-step explanation:

We have been given that in the coordinate plane the vertices of rst are: r (6, -1), s (1, -4), t (-5,6). We are asked to prove that rst is a right triangle.

First of all, we will find the slope between points 'r' and 's' and 's' and 't'.

$\text{Slope}_{rs}=\frac{-1-(-4)}{6-1}=\frac{-1+4}{5}=\frac{3}{5}$

$\text{Slope}_{st}=\frac{-4-6}{1-(-5)}=\frac{-10}{1+5}=-\frac{10}{6}=-\frac{5}{3}$

We know that slope of perpendicular lines in negative reciprocal of each other.

Let us find the product of both slopes.

$\frac{3}{5}\times \frac{-5}{3}=-1$

Since product of both slopes is , therefore, line 'rs' is perpendicular to line 'st' and 'rst' is a right triangle.

Let us assume that coordinates of point p are (x,y) .

We need to find slope of point 'r' to 'p' is equal to slope of point 's' to point 't' and slope of point 'p' to 't' is equal to slope of point 'r' to point 's'.

$\text{Slope}_{rp}=\frac{-1-y}{6-x}=-\frac{5}{3}$

$\text{Slope}_{pt}=\frac{y-6}{x-(-5)}=\frac{3}{5}$

Now, we have two equations and two unknown.

$\frac{y-6}{x+5}=\frac{3}{5}$ and $\frac{-1-y}{6-x}=-\frac{5}{3}$

To solve our system, we will use substitution method.

$y=\frac{3}{5}(x+5)+6$

Upon substituting this value in equation (2), we will get:

$\frac{-1-(\frac{3}{5}(x+5)+6)}{6-x}=-\frac{5}{3}$

$\frac{-\frac{5}{5}-\frac{3}{5}(x+5)-\frac{30}{5}}{6-x}=-\frac{5}{3}$

$\frac{\frac{-5-3x-15-30}{5}}{6-x}=-\frac{5}{3}$

$\frac{-50-3x}{5(6-x)}=-\frac{5}{3}$

Cross multiply:

3(-50-3x)=-25(6-x)

-150-9x=-150+25x

-9x=25x

-9x+9x=25x+9x

34x=0

x=0

Upon substituting x=0 in equation $y=\frac{3}{5}(x+5)+6$ , we will get:

$y=\frac{3}{5}(0+5)+6$

$y=\frac{3}{5}(5)+6$

y=3+6=9

Therefore, the coordinates of point p at (0,9) that will make quadrilateral rstp is a rectangle.

Answer from: Quest

expression #3 is not equivalent. the others are.

Answer from: Quest

the most boring ever written

step-by-step explanation:

i am writing something. yes, i plan to make it the most boring thing ever written. i go to the store. a car is parked. many cars are parked or moving. some are blue. some are tan. they have windows. in the store, there are items for sale. these include such things as soap, detergent, magazines, and lettuce. you can enhance your life with these products. soap can be used for bathing, be it in a bathtub or in a shower. apply the soap to your body and rinse. detergent is used to wash clothes. place your dirty clothes into a washing machine and add some detergent as directed on the box. select the appropriate settings on your washing machine and you should be ready to begin. magazines are stapled reading material made with glossy paper, and they cover a wide variety of topics, ranging from news and politics to business and stock market information. some magazines are concerned with more recreational topics, like sports card collecting or different kinds of hairstyles. lettuce is a vegetable.

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In the coordinate plane the vertices of rst are: r(6 -1), s (1 -4), t (-5,6). Prove that rst is a ri...

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