Finding the coordinates for a missing vertex in a right-angled triangle is a common exercise for high school geometry students. Solving this kind of problem requires an understanding of the Cartesian coordinate grid and basic properties of triangles. Missing coordinates can be found with or without graph paper. When finding the missing coordinates for right triangles, students must be given two coordinate pairs, the area of the larger triangle, and a diagram of the triangle.

Examine the triangle you will be solving. Draw it neatly on a piece of white or graph paper, as it appears in the given diagram. Label any sides that are given in the problem, as well as any coordinates for the vertices of the triangle; in problems of this type, the coordinates of two of the vertices will be given, with one coordinate pair missing. Write down the area of the triangle to the side.

Draw the triangle again, either on a sheet of graph paper or white paper with a rough grid sketch, and plot the given coordinates for the triangle as points. For example, if one of the given coordinates was (2,4), draw a dot that is two tick marks to the right and four tick marks above the origin, or (0,0). Plot the other given point. Draw a line between the two plotted points. Examine the diagram of the triangle provided in the problem and place the third point with unknown coordinates approximately where it should be in relation the other two points. Connect the third point with the other two.

Label the right angle in the triangle that you drew; it should appear between the two legs of the triangle, directly across from the hypotenuse. Write down the equation for the area of a triangle: Area = ½ (length of base leg) * (length of height leg). Plug in the value of the area and either the base or height leg length, depending on which is provided in the problem. Plug in the difference of the x terms, including the variable, if the base is missing, or the difference of the y terms if the height is missing. For example, if the height is unknown, you know that the coordinate of the right angle vertex is (3,5), but you do not know the coordinates at the top of the height segment, plug (y - 5) into the equation for the length of height leg.

Confirm that there is only one unknown variable left in the equation. Solve for the variable, using basic algebraic operations, and write the value to the side. Look back to your right triangle and deduce the other half of the coordinate pair by tracing across from a known point over to the unknown point, either horizontally or vertically depending on the orientation of the triangle. Write your complete coordinate pair in the form (x,y) and circle your answer.

#### Tip

Use a ruler when connecting your points to draw a neat triangle. Neatness in math reduces the chances for confusion.