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.

Examples

Example 1

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

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Worked Solution

Create a strategy

The triangles appear to be right triangles. Use the slope formula to confirm the presence of a right angle, then use the HL congruence theorem.

Apply the idea

For \triangle{ABC}:

Slope of \overline{AB}: \dfrac{-2-1}{0-0}=\text{undefined}

Slope of \overline{AC}: \dfrac{-2+2}{0-4}=0

Since \overline{AB} is vertical and \overline{AC} is horizontal, they are perpendicular and there is a right angle at \angle{A}.

Using HL for congruence, we calculate the lengths of the hypotenuse and a corresponding pair of legs for each triangle:

Hypotenuse \overline{BC} for triangle ABC = \sqrt{\left( 4-0 \right)^2 \left( -2 + 2 \right)^2}=4

Hypotenuse \overline{YZ} for triangle XYZ = \sqrt{\left( -2+2 \right)^2 + \left(2-5 \right)^2}=4

Leg \overline{AB} for triangle ABC = \sqrt{\left(0-0 \right)^2 + \left(-2-1\right)^2}=3

Leg \overline{XY} for triangle XYZ= \sqrt{\left(-2+2 \right)^2 + \left(2-5\right)^2}=3

Both triangles have a right angle, a leg of length 3 units, and a hypotenuse of length 4 units, making them congruent by the HL theorem.

Reflect and check

Suppose we want to prove that the triangles are congruent using an indirect proof, or a proof by contradiction:

Assumption: Assume triangles ABC and XYZ are not congruent.

Based on the coordinate grid, the triangles appear to be right triangles. Let's assume that they are not right triangles to start. By contradiction, we would show that the triangles are in fact right triangles:

For \triangle{ABC}:

Slope of \overline{AB}: \dfrac{-2-1}{0-0}=\text{undefined}

Slope of \overline{AC}: \dfrac{-2+2}{0-4}=0

Since \overline{AB} is vertical and \overline{AC} is horizontal, they are perpendicular.

According to our assumption, this would mean that two triangles having perpendicular sides aren't necessarily right triangles. This is a contradiction to the definition of a right triangle, which states that a right triangle has one angle measuring 90 \degree.

Contradiction with hypotenuse: Given that the hypotenuse \overline{BC} of \triangle{ABC} is 4 units and the hypotenuse \overline{YZ} of triangle \triangle{XYZ} is also 4 units, our assumption would imply that two right triangles with hypotenuses of equal length are not necessarily congruent, which goes against our known properties of right triangles.

Contradiction with leg: Now, the leg \overline{AB} of \triangle{ABC} has a length of 3 units, and the leg \overline{XY} of \triangle{XYZ} also has a length of 3 units. According to our assumption, this would mean that two right triangles with one leg of equal length and hypotenuses of equal length are not necessarily congruent. This is a contradiction to the HL theorem, which states that if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.

Conclusion: Since our assumption led to a contradiction, it must be false. Therefore, \triangle{ABC} and \triangle{XYZ} are congruent by the HL theorem.

Example 2

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

Create a strategy

To prove the triangles are similar, we can use the Side-Side-Side (SSS) similarity criterion by showing that the ratios of the corresponding sides of the two triangles are equal. This involves calculating the lengths of the sides using the distance formula.

Draw the triangles on a coordinate grid and label the vertices.

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Apply the idea

Calculate the lengths of the sides for triangle ABC and triangle PQR, then compare the ratios.

For \triangle ABC:

AB = \sqrt{\left(4-1\right)^2 + \left(6-2\right)^2} = \sqrt{9 + 16} = 5

BC = \sqrt{\left(4-6\right)^2 + \left(6-2\right)^2} = \sqrt{4 + 16} = \sqrt{20}=4\sqrt{5}

CA = \sqrt{\left(6-1\right)^2 + \left(2-2\right)^2} = \sqrt{25} = 5

For \triangle PQR:

PQ = \sqrt{\left(8-2\right)^2 + \left(12-4\right)^2} = \sqrt{36 + 64} = 10

QR = \sqrt{\left(8-12\right)^2 + \left(12-4\right)^2} = \sqrt{16 + 64} = \sqrt{80}=4\sqrt{5}

RP = \sqrt{\left(12-2\right)^2 + \left(4-4\right)^2} = \sqrt{100} = 10

The ratios of the corresponding sides are equal:

\dfrac{AB}{PQ} = \dfrac{5}{10} = \dfrac{1}{2},\,\dfrac{BC}{QR} = \dfrac{2\sqrt{5}}{4\sqrt{5}} = \dfrac{1}{2},\,\dfrac{CA}{RP} = \dfrac{5}{10} = \dfrac{1}{2}

\triangle ABC and \triangle PQR are similar by the SSS similarity criterion.

Reflect and check

When showing two non-right triangles are or are not similar, the Side-Side-Side (SSS) similarity criterion is usually easiest.

However, the AA similarity theorem works as well. This theorem requires two pairs of corresponding angles to be congruent, which we can do by identifying transformations and calculating slopes.

For this example, triangle PQR has been dilated by a factor of 2 where the center of dilation is the origin. This means the orientation of the triangles are the same, so the slopes of corresponding sides will be parallel.

\displaystyle m_{\overline{AB}}	\displaystyle =	\displaystyle \dfrac{4}{3}	Slope of \overline{AB}
\displaystyle m_{\overline{BC}}	\displaystyle =	\displaystyle -2	Slope of \overline{BC}
\displaystyle m_{\overline{CA}}	\displaystyle =	\displaystyle 0	Slope of \overline{CA}
\displaystyle m_{\overline{PQ}}	\displaystyle =	\displaystyle \dfrac{4}{3}	Slope of \overline{PQ}
\displaystyle m_{\overline{PQ}}	\displaystyle =	\displaystyle -2	Slope of \overline{QR}
\displaystyle m_{\overline{RP}}	\displaystyle =	\displaystyle 0	Slope of \overline{RP}

Because the corresponding sides of each triangle are parallel, the angles between them are congruent.

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).

Outcomes

G.TR.2

The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.

G.TR.2c

Use coordinate methods, such as the slope formula and the distance formula, to prove two triangles are congruent.

G.TR.3

The student will, given information in the form of a figure or statement, prove and justify two triangles are similar using direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar triangles.

G.TR.3c

Use coordinate methods, such as the slope formula and the distance formula, to prove two triangles are similar.

11.05 Similarity and congruence in the coordinate plane

Ideas

Similarity and congruence in the coordinate plane