Week 10, Session 2: Coordinate Geometry—Area, Perimeter, and Diagonal Distance on the Grid

Week 10, Session 2: Coordinate Geometry—Area, Perimeter, and Diagonal Distance on the Grid

Self-study purpose

This session combines geometry formulas with coordinates. On GED®-style tasks, a coordinate grid may replace a traditional labeled diagram.

What you should be able to do after this session

• Find side lengths from coordinate differences.
• Compute area and perimeter of grid-aligned rectangles and triangles.
• Use the Pythagorean theorem for diagonal sides.
• Interpret coordinate shapes in real-world layouts.

Visual model

Formula and idea bank

• Horizontal distance: \(|x_2-x_1|\).
• Vertical distance: \(|y_2-y_1|\).
• Rectangle area: \(A=lw\).
• Diagonal distance: \(d=\sqrt{(\Delta x)^2+(\Delta y)^2}\).

Teacher explanation

• For grid-aligned rectangles, count units or subtract coordinates.
• For triangles on a grid, use base and height when they are horizontal/vertical.
• For slanted sides, form a right triangle and use Pythagorean theorem.

Worked example

A rectangle has vertices \((1,2),(6,2),(6,5),(1,5)\). Width is \(6-1=5\), height is \(5-2=3\), so area is \(15\) square units.

GED® problem-solving routine

1. List the coordinates.
2. Find horizontal and vertical lengths.
3. Apply the appropriate formula.
4. For slanted edges, use the distance formula or Pythagorean theorem.

Common mistakes to avoid

• Subtracting coordinates in the wrong direction and keeping a negative length.
• Using diagonal distance as area.
• Counting grid squares incorrectly when coordinates are not drawn.

Mastery check

Before moving to the next session, complete the 10 questions below without notes. Then correct every missed problem by writing: given information → formula → substitution → answer with units.

Advanced Extension and Harder GED® Practice

Deeper idea

Coordinate geometry mixes graph skills with measurement. Axis-aligned shapes are fast because lengths are differences in coordinates. Diagonal sides need the distance formula or Pythagorean theorem.

Worked example: triangle area on a grid

A triangle has vertices \((1,2)\), \((7,2)\), and \((7,8)\).

1. Base from \((1,2)\) to \((7,2)\): \(6\).
2. Height from \((7,2)\) to \((7,8)\): \(6\).
3. Area: \(\frac12(6)(6)=18\).

Harder GED® challenge

A quadrilateral has vertices \((0,0)\), \((6,0)\), \((6,4)\), and \((0,7)\). Split it into a \(6\) by \(4\) rectangle plus a right triangle with base 6 and height 3. Area: \(24+9=33\).

Advanced trap to watch

Do not count grid boxes diagonally. Diagonal distance is longer than horizontal or vertical change and usually needs the Pythagorean theorem.