Mathematics, 11.06.2020 13:57 samy14

Triangle ABC has an angle of 90 degree at B. Point A is on the y-axis, AB is part of the line x - 2y + 8 = 0, and C is the point (6,2). (i) Sketch the triangle.
(ii) Find the equations of the lines AC and BC.
(iii) Find the lengths of AB and BC and hence find the area of the Triangle.
(iv) Using your answer from (iii) find the length of the perpendicular from B to AC

Answers: 3

Show answers

Answers

Answer from: jeffrijv

(i) See attachment for further details, (ii) $y = -2\cdot x +14$ (for the segment BC), $y = -\frac{1}{3} \cdot x + 4$ (for the segment AC), (iii) $l_{AB} \approx 4.472$ , $l_{BC} \approx 4.473$ , A = 10.002 , (iv) $l_{\perp} \approx 3.162$

Step-by-step explanation:

i) The sketched triangle is presented in the figure attached below.

ii) Given that BC is perpendicular to AB and one point is known. It is required to know the slope of the line that contains AB and later, the equation of the line is finally found. The explicit form of the line that contains AB is:

$x - 2\cdot y + 8 = 0$

$2\cdot y = x+8$

$y = \frac{1}{2} \cdot x + 4$

The slope of the line containing BC can be found by the following equivalence:

$m_{BC} = - \frac{1}{m_{AB}}$

$m_{BC} = -\frac{1}{\frac{1}{2} }$

$m_{BC} = -2$

Now, the equation of the line is constructed hereafter:

$y - 2 = -2 \cdot (x-6)$

$y - 2 = -2\cdot x + 12$

$y = -2\cdot x +14$ (for the segment BC)

And given points A (0,4) and C (6,2), equation of the line containing AC is easily found. First, the slope of the function is:

$m_{AC} = \frac{2-4}{6-0}$

$m_{AC} = - \frac{1}{3}$

The y-intercept of the function corresponds to the y-component of point A. Then:

b = 4

Finally, the equation of the line containing AC is:

$y = -\frac{1}{3} \cdot x + 4$ (for the segment AC)

(iii) Length of the segment AC is determined by means of the Pythagorean Theorem:

$l_{AC} = \sqrt{(6-0)^{2}+(2-4)^{2}}$

$l_{AC} \approx 6.325$

Now, the location of point is determined by solving the following system of equations:

$y = \frac{1}{2} \cdot x + 4$ (for the segment AB)

$y = -2\cdot x +14$ (for the segment BC)

The value of x is found after algebraic handling:

$\frac{1}{2}\cdot x + 4 = -2 \cdot x +14$

$\frac{5}{2}\cdot x = 10$

x = 4

And the value of y is:

$y = -2\cdot (4) + 14$

y = 6

The location of point B is (4,6).

The length of the segment AB is:

$l_{AB} = \sqrt{(4-0)^{2}+(6-4)^{2}}$

$l_{AB} \approx 4.472$

And the length of BC is:

$l_{BC} = \sqrt{l_{AC}^{2}-l_{AB}^{2}}$

$l_{BC} = \sqrt{6.325^{2}-4.472^{2}}$

$l_{BC} \approx 4.473$

Lastly, the area of the triangle is:

$A = \frac{1}{2}\cdot l_{AB} \cdot l_{BC}$

$A = \frac{1}{2}\cdot (4.472)\cdot (4.473)$

A = 10.002

(iv) The perpendicular contains point B and crosses the segment AC, the slope of the line equation containing such line segment is found by using the following equivalence:

$m = -\frac{1}{m_{AC}}$

$m = -\frac{1}{\left(-\frac{1}{3}\right) }$

m = 3

Finally, the expression is now constructed:

$y - 6 = 3 \cdot (x-4)$

$y = 6 + 3\cdot (x-4)$

$y = 3\cdot x -6$ (for the segment perpendicular to AC)

The following system of equations must be solved in order to determine the location of the point on AC:

$y = -\frac{1}{3} \cdot x + 4$ (for the segment AC)

$y = 3\cdot x -6$ (for the segment perpendicular to AC)

The value of x is found after algebraic handling:

$-\frac{1}{3}\cdot x + 4 = 3\cdot x - 6$

$\frac{10}{3} \cdot x = 10$

x = 3

And the value of y is:

$y = 3\cdot (3) -6$

y = 3

The location of the point D is (3,3).

Now, the length of the perpendicular is calculated by Pythagorean Theorem:

$l_{\perp} = \sqrt{(3-4)^{2}+(3-6^{2})}$

$l_{\perp} \approx 3.162$

Triangle ABC has an angle of 90 degree at B. Point A is on the y-axis, AB is part of the line x - 2y

Answer from: Quest

The answer to your question is the last one

Answer from: Quest

You need to do x2-x1,y2-y1 and divide it by 2 and you should get (1,-3)

Another question on Mathematics

Mathematics, 21.06.2019 13:00

1. what are the missing reasons in the proof 2. it's in the second picture and 3

Answers: 1

Answer

Mathematics, 21.06.2019 20:20

One of every 20 customers reports poor customer service on your company’s customer satisfaction survey. you have just created a new process that should cut the number of poor customer service complaints in half. what percentage of customers would you expect to report poor service after this process is implemented? 1.) 5% 2.) 10% 3.) 2% 4.) 2.5%

Answers: 1

Answer

Mathematics, 21.06.2019 21:50

Prism m and pyramid n have the same base area and the same height. cylinder p and prism q have the same height and the same base perimeter. cone z has the same base area as cylinder y, but its height is three times the height of cylinder y. the figures and have the same volume.

Answers: 3

Answer

Mathematics, 21.06.2019 23:40

30 ! the graph shows the number of siblings different students have. select from the drop-down menus to correctly complete the statement.

Answers: 1

Answer

You know the right answer?

Triangle ABC has an angle of 90 degree at B. Point A is on the y-axis, AB is part of the line x - 2y...

Questions

Mathematics, 26.08.2021 18:50

Are integers rational numbers? Always Sometimes Never...