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

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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

I'm sure the answer is a

Answer from: Quest

unless you made a typo, the answers are both a and c.

step-by-step explanation:

1.5/3 is .5 whereas 3/9 is .3333333 not proportional

22/33 is .66666 and 4/6 is also .66666 proportional

2/4 is .5 whereas 142/213 is .66666 not proportional

14/7 is 2 and 7/3.5 is also 2 proportional

you could've also checked it by doing this simple division:

if the larger fraction divides with the smaller fraction to equal 1, it is proportional. so let me show you what i mean:

when i divide 22/33 by 4/6 i get 1, so we know it is proportional, and it is the same with 14/7 and 7/3.5. now if i divide 3/9 by 1.5/3 i get .66666 so therefore we know it is not proportional.

there are many ways to do this so i wanted to show you some possible ways.

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