Similarity and congruency theorems are useful in proving whether two triangles are similar or similar and congruent.

Remember the similarity theorems for triangles:

Side-Side-Side (SSS\sim)

Side-Angle-Side (SAS\sim)

Angle-Angle (AA)

Remember the congruency theorems for triangles:

SSS

SAS

Angle-Side-Angle (ASA)

Angle-Angle-Side (AAS)

Hypotenuse-Leg (HL)

We may now see triangles on a coordinate grid without given side lengths and angle measures, where tools such as the distance formula and the slope formula help prove similarity or congruence between them.

\displaystyle d=\sqrt{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2}

\bm{d}

the distance between two points

\bm{x_1, x_2}

the x-coordinates of \left(x_1, y_1 \right) and \left(x_2, y_2 \right)

\bm{y_1, y_2}

the y-coordinates of \left(x_1, y_1 \right) and \left(x_2, y_2 \right)

\displaystyle m=\dfrac{y_2 - y_1}{x_2 - x_1}

\bm{m}

the slope between two points

\bm{x_1, x_2}

the x-coordinates of \left(x_1, y_1 \right) and \left(x_2, y_2 \right)

\bm{y_1, y_2}

the y-coordinates of \left(x_1, y_1 \right) and \left(x_2, y_2 \right)

To write a **direct proof** of a theorem using the coordinate plane, we can follow the steps below:

Represent the given information with a labeled diagram on the coordinate plane.

Use the coordinates of the key points to determine other properties of the diagram.

Use formulas like the distance and slope formulas on the coordinate grid to prove the theorem, depending on what's required (e.g., side lengths, right angles).

An **indirect proof** for triangle congruence might involve assuming two triangles are not congruent despite having three pairs of congruent sides, and then showing this assumption leads to a contradiction.

Prove that triangle \triangle{ABC} \cong \triangle{XYZ}.

Worked Solution

Triangle ABC has vertices at A \left(1,2 \right), B \left(4,6 \right), and C \left(6,2\right). Triangle PQR has vertices at P \left(2,4 \right), Q \left(8,12 \right), and R \left(12,4 \right). Prove that \triangle{ABC} \sim \triangle{PQR}.

Worked Solution

Idea summary

To write a **direct proof** of a theorem using the coordinate plane, we can follow the steps below:

Represent the given information with a labeled diagram on the coordinate plane.

Use the coordinates of the key points to determine other properties of the diagram.

Use formulas like the distance and slope formulas on the coordinate grid to prove the theorem, depending on what's required (e.g., side lengths, right angles).