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The Area of triangles 2 exercise appears under the 6th grade (U.S.) Math Mission and High school geometry Math Mission. This exercise practices finding the area of triangles that are not set up in a standard position.

## Types of Problems

There is one type of problem in this exercise:

1. Find the area of the triangle on the grid: This problem provides a triangle placed on a rectangular grid with all vertices sitting at lattice points. Most of the time, however, none of the sides are parallel to the directions of the grid. The user is asked to find the area of the triangle and write the answer in the space provided.

## Strategies

Knowledge of the area of a triangle formula and an ability to complete a triangle into a rectangle are needed on this problem. It is also possible to do this problem using distance formula plus Heron's formula, but why?

1. The formula for the area of a triangle is base times height divided by two, or one-half of base times height.
2. The triangle can be "completed" into a rectangle by inserting several right triangles in the corners. Then the area of the triangle itself can be found by subtracting the area of the right triangles from the overall rectangle.
3. The formula for the area of a triangle is ${\frac{(b\times h)}{2}}$, or ${\frac{1}{2}\times (b\times h)}$.
4. A programmable TI-83 or TI-84 calculator can be used to write a program to do this problem. This code uses Heron's Formula to find the area:
:ClrHome
:Prompt A,B,C,D,E,G
:√((A-C)^2 + (B-D)^2) -> X
:√((A-E)^2 + (B-G)^2) -> Y
:√((B-E)^2 + (C-G)^2) -> Z
:(X+Y+Z)/2 -> S
:√(S*(S-X)*(S-Y)*(S-Z))
:Disp Ans

## Real-life Applications

1. Area and perimeter are used in many jobs for architecture and interior design.
2. Architects use triangles when building bridges, roofs on houses, and other structures.
3. The ancient Egyptians from over 4000 years ago were very good at shapes and geometry. Every time the Nile burst its banks and flooded the planes, they had to use geometry to measure their gardens and fields all over again.