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Solve geometric problems using analytic geometry

Geometric relationships are represented and solved in the Cartesian plane using coordinates, slope, distance, midpoint, and equations of lines. Algebraic conditions such as collinearity, equal lengths, parallelism, perpendicularity, and line intersection are interpreted as geometric properties to determine unknown coordinates, lengths, and classifications of triangles and quadrilaterals, supporting coordinate proofs. The scope is limited to linear coordinate geometry and standard distance and midpoint formulas, excluding conic sections, parametric equations, vectors, and more advanced coordinate methods.

Detailed Explanation: Solve geometric problems using analytic geometry

Use coordinates to turn geometric facts into calculations. For a triangle, you can:

  • use slope to test whether sides are parallel or perpendicular;
  • use the distance formula to compare side lengths.

Example

Classify the triangle with vertices

A(1,2),B(5,2),C(5,5).A(1,2), \qquad B(5,2), \qquad C(5,5).

Step 1: Find the slopes

The slope between (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) is

m=y2−y1x2−x1.m=\frac{y_2-y_1}{x_2-x_1}.

For AB‾\overline{AB}:

mAB=2−25−1=04=0.m_{AB}=\frac{2-2}{5-1}=\frac{0}{4}=0.

So AB‾\overline{AB} is horizontal.

For BC‾\overline{BC}:

mBC=5−25−5=30.m_{BC}=\frac{5-2}{5-5}=\frac{3}{0}.

The slope is undefined, so BC‾\overline{BC} is vertical.

A horizontal line and a vertical line are perpendicular. Therefore,

∠B=90∘.\angle B=90^\circ.

The triangle is a right triangle.

Step 2: Find the side lengths

The distance formula is

d=(x2−x1)2+(y2−y1)2.d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.

For AB‾\overline{AB}:

AB=(5−1)2+(2−2)2=16=4.AB=\sqrt{(5-1)^2+(2-2)^2} =\sqrt{16}=4.

For BC‾\overline{BC}:

BC=(5−5)2+(5−2)2=9=3.BC=\sqrt{(5-5)^2+(5-2)^2} =\sqrt{9}=3.

For AC‾\overline{AC}:

AC=(5−1)2+(5−2)2=16+9=25=5.AC=\sqrt{(5-1)^2+(5-2)^2} =\sqrt{16+9} =\sqrt{25}=5.

All three side lengths are different, so the triangle is scalene.

Therefore, the triangle is a right scalene triangle, with the right angle at BB.

Learn by doing: Solve geometric problems using analytic geometry

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