AccipiterQ wrote:
The (x, y) coordinates of points P and Q are (-2, 9) and (-7, -3), respectively. The height of equilateral triangle XYZ is the same as the length of line segment PQ. What is the area of triangle XYZ?
A) \((169*\sqrt{3})/3\)
B) 84.5
C) 75*\(\sqrt{3}\)
D) \((169*\sqrt{3})/4\)
E) \((225*\sqrt{3})/4\)
The formula for the distance between two points (x1, y1) and (x2, y2) is:
.
One way to understand this formula is to understand that the distance between any two points on the coordinate plane is equal to the hypotenuse of a right triangle whose legs are the difference of the x-values and the difference of the y-values (see figure). The difference of the x-values of P and Q is 5 and the difference of the y-values is 12. The hypotenuse must be 13 because these leg values are part of the known right triangle triple: 5, 12, 13.
We are told that this length (13) is equal to the height of the equilateral triangle XYZ. An equilateral triangle can be cut into two 30-60-90 triangles, where the height of the equilateral triangle is equal to the long leg of each 30-60-90 triangle. We know that the height of XYZ is 13 so the long leg of each 30-60-90 triangle is equal to 13. Using the ratio of the sides of a 30-60-90 triangle (1:\(\sqrt{3}\): 2), we can determine that the length of the short leg of each 30-60-90 triangle is equal to 13/. The short leg of each 30-60-90 triangle is equal to half of the base of equilateral triangle XYZ. Thus the base of XYZ = 2(13/) = 26/.
The question asks for the area of XYZ, which is equal to 1/2 × base × height:
The correct answer is A.
For the LIFE of me I don't get why the OE is correct, and why my method is wrong, I'll post my method in the first reply
OK, so the OE states to figure out what PQ equals, which is very easy, just pythagoras it, and you get 13. It then says that this number is equal to the height of equilateral triangle XYZ. So my logic is this:
we know that the height of an equilateral triangle = \(\sqrt{3}/2\)*side. So since we already know the height, we can solve for any side:
13= \(\sqrt{3}/2\)*side
26=\(\sqrt{3}\)*side
\(26/\sqrt{3}\)=side
Great, now we have a side.
Area of an equilateral triangle? \((side^2*\sqrt{3})/4\)
so we have \((26/\sqrt{3}^2)*\sqrt{3}\))/4
that ends up with \(((676/3)*\sqrt{3}\))/4 which is obviously not the right answer. How on Earth does this not work??
edit: edited for formatting