*find eq of perpendicular line so that the area of the triangle is 8

[karush]

View attachment 3268
for the line y=4x-2 there is one perpendicular line of which will enclose a triangle on the lines and the values of the y-axis whose area is 8. What is the equation of this line?

Well, I chose the x value to be the height of the triangle and that would make the base \(\displaystyle B=(4x-2)+2+\frac{1}{4}x\) or just $B=\frac{17}{4}h$ if $h=$ the $x$ value.

just seeing if I am going the right direction with this seem more complicated than is should be. got to be a slam dunk method...

I got a weird answer for $h=\frac{8}{\sqrt{17}}$ and $b=2\sqrt{17}$

 

[mathmari]

View attachment 3268
for the line y=4x-2 there is one perpendicular line of which will enclose a triangle on the lines and the values of the y-axis whose area is 8. What is the equation of this line?

Well, I chose the x value to be the height of the triangle and that would make the base \(\displaystyle B=(4x-2)+2+\frac{1}{4}x\) or just $B=\frac{17}{4}h$ if $h=$ the $x$ value.

just seeing if I am going the right direction with this seem more complicated than is should be. got to be a slam dunk method...

I got a weird answer for $h=\frac{8}{\sqrt{17}}$ and $b=2\sqrt{17}$

The area of a triangle is given from the formula $$\text{ Area }=\frac{1}{2} B H$$

You will find the part of the basis $B$ that is over the $x$-axis, by setting at the equation $y=\frac{x}{4}+b$, $x=0$. So, it is equal to $b$.
The part of $B$ under the $x$-axis is equal to $2$.

Therefore, $B=b+2$.

You will find the height $H$ by finding the intersection between the two line equations:
$$\frac{x}{4}+b=4x-2 \Rightarrow x=\frac{4b+2}{15}$$

Therefore, $H=\frac{4b+2}{15}$.

So, to calculate $b$ we have to solve the following:

$$B \cdot H=8 \Rightarrow \frac{b+2}{2} \frac{4b+2}{15}=8$$

 

[johng1]

Karush,
Actually, there are 2 such triangles.

 

[karush]

ok I posted this problem also on Linkedin since it had over 1000 views
but there was an image with this problem which was from a SAT pdf
but I couldn't find it but apparently the image is not necessary to solve it
Anyway

 

[CellCoree]

which would make the equation of the perpendicular line $y=-\frac{\sqrt{17}}{4}x+\frac{8}{\sqrt{17}}$.

It looks like you are on the right track! To find the equation of the perpendicular line, we can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. In this case, the slope of the given line is 4, so the slope of the perpendicular line would be -1/4.

Now, we can use the point-slope form of a line to find the equation. We know that the line passes through the point (0, -2) since it intersects the y-axis at -2. So, the equation would be y - (-2) = -1/4(x-0), which simplifies to y = -1/4x - 2.

To find the base of the triangle, we can use the distance formula between two points. Using the points (0, -2) and (-4, 0) (since the x-value of the perpendicular line is -4), we get a base length of 2√17.

Finally, we can use the formula for the area of a triangle, A = 1/2 * base * height, with the given area of 8 and the base of 2√17 to solve for the height. This gives us a height of 8/√17.

Putting it all together, the equation of the perpendicular line would be y = -1/4x + 8/√17. We can simplify this by multiplying the numerator and denominator of the fraction by √17, giving us y = -√17/4x + 8. This is equivalent to the equation you found, just with the fraction simplified. Great job!