All categories

2 years ago

Help please! how do I calculate the coordinates of point a so that at is an altitude?

Mathematics

1 answer:

vodomira [7]2 years ago

6 0

Part (a)

Since we want segment AT to be an altitude, this means that point A is somewhere on segment RI such that angle TAI is 90 degrees. In other words, segments AT and RI are perpendicular. Check out the diagram below.

We'll need to find the slope of line RI, so we'll need the slope formula

R = (x1,y1) = (-2,-2)

I = (x2,y2) = (4,1)

m = slope of line RI

m = (y2-y1)/(x2-x1)

m = (1-(-2))/(4-(-2))

m = (1+2)/(4+2)

m = 3/6

m = 1/2

m = 0.5

The slope of line RI is 1/2 or 0.5

Now let's use point R(-2,-2) along with that slope we just found to find the equation of line RI.

Turn to point slope form

y - y1 = m(x - x1)

y - (-2) = 0.5(x -(-2))

y + 2 = 0.5(x + 2)

y + 2 = 0.5x + 1

y = 0.5x + 1 - 2

y = 0.5x - 1

This is the equation of line RI. We'll use it later.

----------

We found that the slope of line RI was 1/2.

The negative reciprocal involves us flipping the fraction and flipping the sign to get -2/1 or simply -2.

The perpendicular slope is -2. This is the slope of altitude AT.

We want line AT to go through T(-2,4), so we'll use this point along with the perpendicular slope we just found to calculate the equation of line AT.

We'll turn to the point slope form

y - y1 = m(x - x1)

y - 4 = -2(x - (-2))

y - 4 = -2(x + 2)

y - 4 = -2x - 4

y = -2x - 4 + 4

y = -2x

The equation of the line that goes through points A and T is y = -2x.

----------

To summarize so far, we have these two equations

y = 0.5x - 1 .... equation of line RI
y = -2x .... equation of line AT

Apply substitution to solve for (x,y) to find the intersection point. This will determine where point A is located.

y = 0.5x - 1

-2x = 0.5x - 1

-2x-0.5x = -1

-2.5x = -1

x = -1/(-2.5)

x = 0.4

Then we can determine y like so

y = -2x

y = -2(0.4)

y = -0.8

Therefore, point A is located at (0.4, -0.8). Both coordinate values are exact. In fraction form, we can say that the point is (2/5, -4/5), but I find decimals easier to work with in this instance.

<h3>Answer: (0.4, -0.8)</h3>

===============================================================

Part (b)

To find the length of segment AT, we can find the distance from A to T. As you can probably guess, we'll use the distance formula.

A = (x1,y1) = (0.4, -0.8) ... found back in part (a)

T = (x2,y2) = (-2, 4) ... given

$d = \sqrt{ (x_2-x_1)^2+(y_2-y_1)^2 }\\\\d = \sqrt{ (-2-0.4)^2+(4 - (-0.8))^2 }\\\\d = \sqrt{ (-2-0.4)^2+(4 + 0.8)^2 }\\\\d = \sqrt{ (-2.4)^2+(4.8)^2 }\\\\d = \sqrt{ 5.76+23.04 }\\\\d = \sqrt{ 28.8 }\\\\d \approx 5.36656\\\\d \approx 5.37\\\\$

The distance from point A to point T is approximately 5.37 units. This means that segment AT is roughly 5.37 units long.

<h3>Answer: 5.37 units</h3>

You might be interested in

HELPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

Gelneren [198K]

Answer:

Step-by-step explanation:

Hurley drank between 4 and 5 liters of water for 8 different days and drank between 1 or 2 liters of water on 4 different days. To find the answer we have to use the following equation.

# of days Hurley drank (4-5 L) - # of days Hurley drank (1-2 L) = # of more days

8 - 4 = 4

4 0

2 years ago

Alan has read 10% of a book. He has 81 more pages to finish. How many pages are there in the book?

rewona [7]

Answer:

Step-by-step explanation:

100%-10%=90%

90%->81

10%->9

100%->9+81=90

6 0

2 years ago

Surface area in terms of pi?

Lunna [17]

$\bf \textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\ \cline{1-1} V=972\pi \end{cases}\implies 972\pi =\cfrac{4\pi r^3}{3}\implies 2916\pi =4\pi r^3 \\\\\\ \cfrac{2916}{4\pi }=r^3\implies 729=r^3\implies \sqrt[3]{729}=r\implies 9=r \\\\[-0.35em] ~\dotfill\\\\ \textit{surface area of a sphere}\\\\ SA=4\pi r^2\qquad \qquad \implies SA=4\pi (9)^2\implies \boxed{SA=324\pi }$

5 0

2 years ago

On a 20 question quiz that Mrs. MacLeod gave. One student got 13 wrong. What is the percent that student received on the test? &

Alexxx [7]

13/20 = 0.65

0.65 x 100 = 65

They got a 65.

7 0

2 years ago

Suppose m is in the line given by the equation 6x-3y=7, and suppose n is the line perpendicular to m and passing Nd through the

stealth61 [152]

Answer:

The x co-ordinate o intersection of line k and n is $\frac{8}{11}$

Step-by-step explanation:

Given as :

The equation of line m is 6 x - 3 y = 7

So, in the standard form , line equation is

y = a x + c , where a is the slope

So, 6 x - 3 y = 7 can be written as

3 y = 6 x - 7

or, y = 2 x - $\frac{7}{3}$ ........1

So, slope of this line = a = 2

Now, The line n is perpendicular to line m and passing through line ( 6 , 2 )

So, Slope of line n = b

For , perpendicular lines , products of slope = - 1

Or, a × b = -1

∴ b = - $\frac{1}{a}$

I.e b = - $\frac{1}{2}$

So,equation of line n with slope b and passing through line ( 6 , 2 ) is

y - $y_1$ = b ( x - $x_1$ )

or, y - 2 = - $\frac{1}{2}$ ( x - 6 )

or, 2 × ( y - 2 ) = - 1 ( x - 6 )

or, 2 y - 4 = - x + 6

or, x + 2 y -10 = 0 ........2

Again, equation of line k with slope 5 and y intercept = 1

For y intercept , x coordinate = 0

y = c x + c

or, 1 = c× ( 0 ) + c

Or, c = 1

Or, equation of line k is

y = 5 x + 1 ..........3

Now intersection of line k and n is

put the value of y from eq 3 into eq 2

I.e x + 2 × ( 5 x + 1 )-10 = 0

Or, x + 10 x + 2 - 10 = 0

or, 11 x - 8 = 0

or 11 x = 8

∴ x = $\frac{8}{11}$

Hence The x co-ordinate o intersection of line k and n is $\frac{8}{11}$ Answer

4 0

2 years ago