What is the area of the triangle in the diagram? (x 1, y1), (x2,y2), and (0,0)

We can consider the case where (x1, y1) and (x2, y2) are in the first quadrant. We will assume that, together with (0, 0), they are in order clockwise around the triangle.

Consider the lines x=x1, x=x2, the x-axis, and the segment (x1, y1) to (x2, y2). These will form a trapezoid whose area is

... (area of trapezoid) = (1/2)(y1 + y2)(x2 - x1)

(If x2 < x1, the area will be negative.)

LIkewise, the triangle formed by the segment (0, 0) to (x1, y1), the line x=x1, and the x-axis will have area

... (area of first triangle) = (1/2)(y1)(x1)

The triangle formed in similar fashion by the segment (0, 0) to (x2, y2), the line x=x2, and the x-axis will have area

... (area of second triange) = (1/2)(y2)(x2)

You can convince yourself that the total area of the triangle of interest is equal to the sum of the areas computed here:

... (area of triangle in the diagram) = (area of first triangle) + (area of trapezoid) - (area of second triangle)

Note that we have subtracted the area of the second triangle. Then, in terms of coordinates, we have

... (area of triangle in the diagram) = (1/2)(x1·y1 + (y1+y2)(x2-x1) - x2·y2)

... ... = (1/2)(x2·y1 - x1·y2)

If the points are in counterclockwise order, the area will be the negative of this. Thus, we can say ...

... triangle area = (1/2)·|x1·y2 - x2·y1|

This formula will apply whether the points are all in the first quadrant or not, as long as one of them is at the origin.

the clues that signal the reader should change tone are the punctuation marks, the grammatical signs. for example, the quotation marks at the beginning of something someone else said literally or the exclamation marks.

the tone of the first line of dialogue until "gettysburg" is a kind tone, a tone of advise. the narrator is trying to the other person in doing something he or she obviously is finding hard to do by giving a piece of advise and bringing up a memory of a successful similar case.

the clue that the reader understand how to read the word "bang" is the exclamation mark. it gives the word a surprise tone, a strong accent.

the best tone for reading the word "bang" is an exciting tone, a surprise one, even a loud one.

the words that should be read with a formal tone are the ones that give factual information. the sentence: mister lincoln couldn't think of anything to say at the gettysburg" gives information about an event and it needs to be read formally, also, when the narrator wants to transmit calmness, a formal and slow tone is needed, because people also transmit messages with the vibrations of our voices and tones.