# Distance formula vs similar triangles

1. Jun 20, 2017

### fayan77

Hello, took a year off school, now shaking the rust off. so according to the book using similar triangles d=9.6 I understand how they got the answer, but i used distance formula from the point to the origin and got 9.8 I checked 9.6 and it checks out with the numbers but Idk why using the distance formula yields a different answer. am I doing something wrong?

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2. Jun 20, 2017

### scottdave

To use the distance formula, you need the coordinates of that point. How did you find the coordinates? What were the steps that you used?

3. Jun 20, 2017

### fayan77

the book shows point A (4,9) sorry for the picture quality, my phone sucks.

4. Jun 20, 2017

### Staff: Mentor

The picture quality is really problematic.

(4,9) are not the coordinates of any interesting point in the sketch.

9.6 is correct.

5. Jun 20, 2017

### scottdave

(4,9) is a point on that line between (0,12) and (16,0), but it is not the point of intersection {it does not satisfy the (4/3) slope from the origin}. You can find the intersection point by finding where the two lines meet.
It is not hard, just two straight lines to find the (x,y) coordinate. The actual intersection point is (5.76,7.68) which you can now use the distance formula and get the same answer.

6. Jun 20, 2017

### Staff: Mentor

I wouldn't solve this via similar triangles, by the way. The area of a triangle can be calculated as length*height/2. You can use the vertical and horizontal sides as length and height to find the area. Then you can use the long diagonal and the unknown length (also a length/height pair) to find the unknown length.

7. Jun 20, 2017

### fayan77

yeah that is what i did i took the slope and line then derived an orthogonal line passing through origin and found the point. That point on the book is confusing tho. thanks.

8. Jun 20, 2017

### Staff: Mentor

That's not what I meant.

12=3*4 and 16=4*4, and you should recognize 3,4,5 as right triangle. Therefore, the hypothenuse has a length of 5*4=20.

The area of the triangle is 12*16/2.

If we call the unknown length h, we get 12*16/2 = 20*h/2.
The unknown length is then h=12*16/20, and that is something you can calculate easily.

9. Jun 21, 2017

### Ray Vickson

Wouldn't similar triangles be the easiest possible way? Solving $16/20 = d/12$ is about as simple as it gets.

Of course, one would first need to recognize that the large triangle has hypotenuse 20, as you have indicated.

10. Jun 21, 2017

### Staff: Mentor

Well, it leads to the same formula, but you have to think which side corresponds to what.