Question

In a Cartesian coordinate system, the coordinates of two points P and Qare (2,4,4) and (-2,-3,7) respectively. Find the PQ and it's magnitude.

Answer 1

The formula to calculate the distance between two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is d=(x1−x2)2+(y1−y2)2−−−−−−−−−−−−−−−−−−−√d=(x1−x2)2+(y1−y2)2.

So the distance between P and Q is d=(−2+7)2+(9+3)2−−−−−−−−−−−−−−−−√=13d=(−2+7)2+(9+3)2=13. (Else you can find the length of PQ by realizing that PQ is a hypotenuse of 5:12:13 right triangle)

So we know that the height in equilateral triangle XYZ equals to 13: height=13height=13

Now, since the height of the equilateral triangle divides it into two 30-60-90 right triangles with the ratio of the sides 1:3√:21:3:2 then height becomes the leg opposite 60 degrees angle and the hypotenuse, which is the side of an equilateral triangle can be found from the ratio: heightside=3√2heightside=32 --> side=263√side=263.

Next, areaequilateral=side2∗3√4=1693√areaequilateral=side2∗34=1693. (Else, areaequilateral=12∗height∗side=12∗13∗263√=1693√areaequilateral=12∗height∗side=12∗13∗263=1693)

Answer: A.