# Math Help - easy equation problem

1. ## easy equation problem

not good at math so if you could please help it would be great.

The sides of a triangle have these equations:
y= -0.5x+1
y= 2x-4
y=-3x-9
Verify that the triangle is an isosceles right triangle (without graphing)

I've noticed that the second equation is -4 times the amount of the first if that helps

2. Originally Posted by Fox25
not good at math so if you could please help it would be great.

The sides of a triangle have these equations:
y= -0.5x+1
y= 2x-4
y=-3x-9
Verify that the triangle is an isosceles right triangle (without graphing)

I've noticed that the second equation is -4 times the amount of the first if that helps
find the points of intersection (x,y) to locate the vertices of the triangle, then find the distance between them using the distance formula.

two distances should be equal, and all three distances should work in the Pythagorean Thm.

3. thank you skeeter

4. Greetings
The sides of a triangle have these equations:
$y_1= -0.5x+1$
$y_2= 2x-4$
$y_3=-3x-9$

First things first, read over your notes, you wouldn't believe how much that helps.

Now that you've done this (theoretically),let's tackle the question.
To be an isosceles triangle, you need one line perpendicular to another and two lines to be equal to each other in length.

In coordinate geometry, when one line is perpendicular to another, the gradient of the line is the negative reciprocal of the other.
I.E. When the 2 gradients are multiplied, the product is negative 1.

You may see now that $y_1$ gradient is the negative reciprocal of $y_2$, hence $y_1$ is perpendicular to $y_2$
Hence the triangle formed by the intersection of the 3 lines is right-angled.

Now, all we have to do is determine whether $y_1$ and $y_2$ are equal to each other. $y_3$ is opposite to the right angle, so it's the hypotenuse.

Find where $y_1$ and $y_2$ intersect each other and $y_3$. Now with these coordinates the length of $y_1$ and $y_2$.

$y_1= -0.5x+1$
$y_2= 2x-4$
$y_3=-3x-9$

$-0.5x+1=3x-9$
$x=\frac{20}{7}$
Then $y=\frac{-3}{7}$
$A(\frac{20}{7},\frac{-3}{7})$

$-0.5x+1=2x-4$
$x=2$
Then $y=0$
$B(2,0)$

$2x-4=3x-9$
$x=5$
Then $y=6$
$C(5,6)$

You may graph at this point just to get your visual bearings.

Now just apply the length of a line formula to determine that two of the lines are equal to each other.

Edit: Be more like Skeeter and make it short