I love these. It's often called the Shoelace Formula. It actually works for the area of any 2D polygon.
We can derive it by first imagining our triangle in the first quadrant, one vertex at the origin, one at (a,b), one at (c,d), with (0,0),(a,b),(c,d) in counterclockwise order.
Our triangle is inscribed in the 
rectangle. There are three right triangles in that rectangle that aren't part of our triangle. When we subtract the area of the right triangles from the area of the rectangle we're left with the area S of our triangle.
That's the cross product in the purest form. When we're away from the origin, a arbitrary triangle with vertices 
will have the same area as one whose vertex C is translated to the origin.
We set 
That's a perfectly useful formula right there. But it's usually multiplied out:
That's the usual form, the sum of cross products. Let's line up our numbers to make it easier.
(1, 2), (3, 4), (−7, 7)
(−7, 7),(1, 2), (3, 4),
[tex]A = \frac 1 2 ( 1(7)-2(-7) + 3(2)-4(1) + -7(4) - (7)(3)
Answer:
She runs for 1.6 hours
Step-by-step explanation:
For this, one way to do it is to just set 1 hour over 4 miles equal to x hours for 6.4 miles. The reason this is done is because if the speed is constant that means the rate stays the same, which means the ratio of miles to hours should always be the same, so the two ratios must be equivalent.
1/4 = x/6.4
Multiply both sides by 6.4 to get x by itself:
6.4/4 = x
1.6 = x
Hope that helps!
Answer:
379,442 AU
Step-by-step explanation:
The angle bisector is the altitude of the triangle. The resulting right triangle has a distance of 1 AU opposite the 0.000151° angle. The length we're interested in is the hypotenuse of the triangle. The relevant trig relation is ...
Sin = Opposite/Hypotenuse
Hypotenuse = Opposite/Sin
E-S distance = (1 AU)/sin(0.000151°) ≈ 379,442 AU
The star is about 379,442 AU from Earth.
Answer:
x * 17.5 = y
Step-by-step explanation:
Answer:
1: 12 and 48
2: 45 and 15
3: 2 and 58
Step-by-step explanation:
60-12=38
60-15=45
60-2= 58
