Why Is This Triangle ALWAYS Half the Rectangle?

Step by step

1. 12 ft × 8 ft garden · the top can be anywhere · triangle?
   This garden is twelve feet by eight. The top of this triangle can be anywhere on the far side. How big is the triangle?
2. Always 48 sq ft = half · "the half model"
   It's always forty-eight square feet: exactly half the garden. In China, this is called the half model.
3. Way 1: a line straight down → two rectangles
   Way one: drop a line straight down from the top corner. It cuts the garden into two smaller rectangles.
4. Each side = a diagonal → cuts its rectangle in half
   Each side of the triangle is a diagonal of one small rectangle. And a diagonal cuts a rectangle exactly in half.
5. Half + half → half the garden
   So the triangle is half of the left part, plus half of the right part. That's half of the whole garden!
6. Way 2: 12 × 8 ÷ 2 = 48 = half of 12 × 8
   Way two is the school formula. Twelve times eight, divided by two, is forty-eight. The garden is twelve times eight, so the triangle is exactly half.
7. Bonus: a point inside → four triangles
   Here's a bonus. Put a point anywhere inside, and join it to all four corners. That makes four triangles.
8. A line across through the point · look at the top strip
   Draw a line through the point, straight across. Now look only at the top strip.
9. Base on one side, point on the other → half of the strip
   The top triangle has its base on one side of this strip, and its point on the other side. That's our half model again. Watch the outside pieces turn and fill it.
10. Bottom triangle = half of the bottom strip
    The bottom strip works the same way. The bottom triangle is half of the bottom strip.
11. Half + half → top + bottom = half the rectangle
    Half of the top strip, plus half of the bottom strip, is half of the whole rectangle. So top plus bottom is half!
12. Line the other way → left + right = the other half
    Turn the line the other way, and the left and right triangles work the same. They make the other half.
13. Base on one side, top on the other → half
    So a triangle with its base on one side and its top on the opposite side is always half the rectangle.
14. On a test: base 12 · height 8 · 12 × 8 ÷ 2 = 48 sq ft
    On a test, write it like this. The base is the near side: twelve feet. The top is on the far side, so the height is eight feet. Base times height, divided by two, is forty-eight. The triangle is forty-eight square feet: half the garden.