Right Angle's Vertex Is On The Line!

Algebra Level 5

Suppose we have two points $A(5, 5)$ and $B(3, 2)$ . Point C is placed on the line $y = 3x+4$ such that $\triangle ABC$ is a right triangle.

If $C$ has coordinates $(x, y)$ and the sum of all possible values of $(x, y)$ is $P$ , then find $11P$ .

The answer is 140.

1 solution

Trevor Arashiro
Oct 3, 2014

I figured a pretty nice solution for this. The main reason I made this problem was to show this neat solution. Btw, I'll add a picture when I get a chance, but for now I gotta get on a plane.

First we begin by finding the slope between the two points a,b which is 3/2. Now, to create a perpendicular to this, we need a line with a slope of -2/3. This will cause the 90 to occur at angles a,b.

Now, here's the cool part. Rather than finding two intersections that the Lines running through A and B make with the line $y=3x+4$ , we find the mid point of $\overline{AB}$ and the intersection its line makes.

The mid point of $\overline{AB}$ is (4,7/2). Thus using point slope form, we get

$y-4=-\frac{2}{3}(x-\frac{7}{2})$

$y=-\frac{2x}{3}+\frac{37}{6}$

next, we set this equal to $y=3x+4$

$3x+4=-\frac{2x}{3}+\frac{37}{6}$

$(\frac{13}{22},\frac{127}{22})$ .

Finally, we multiply this by 2 since it's the average of the two points and by 11 since the question asks for 11P.

$13+127=140$ .

Y did u consider a perpendicular through the midpoint of AB?As far as I understand, its not necessary for a arbitrary line to intersect a perpendicular such that the point obtained would make a right angled triangle at that point. Please explain :-)

Jai Lodha - 6 years, 8 months ago

If it is a perpendicular, the it is guaranteed that you will get a right angle. I will add a picture now

Trevor Arashiro - 6 years, 8 months ago

Right Angle's Vertex Is On The Line!

The answer is 140.

1 solution

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