Geometric Twins | FIO

Question 17

$\text{ABCD}$ is a square. Show that $\Delta \text{ABC} \cong \Delta \text{ADC}$ . Is $\Delta \text{ABC}$ also congruent to $\Delta \text{CDA}$ ?

Give more examples of two triangles where one triangle is congruent to the other in two different ways, as in the case above. Can you give an example of two triangles where one is congruent to the other in six different ways?

Solution

Understand the Question

A square $\text{ABCD}$ has all four sides equal in length: $\text{AB} = \text{BC} = \text{CD} = \text{DA}$ .
Two triangles are congruent by the SSS (Side-Side-Side) criterion if all three pairs of corresponding sides are equal.
If a triangle has symmetries (such as an isosceles or equilateral triangle), its vertices can correspond to another congruent triangle's vertices in more than one way:
- Isosceles triangles: $2$ different correspondences.
- Equilateral triangles: all $3$ sides and angles are equal, giving $3! = 6$ different correspondences.

(i) $\text{ABCD}$ is a square. Show that $\Delta \text{ABC} \cong \Delta \text{ADC}$ .

Step 1 · Prove Congruence of $\Delta \text{ABC}$ and $\Delta \text{ADC}$

Consider square $\text{ABCD}$ with diagonal $\text{AC}$ . Diagram 1

In $\Delta \text{ABC}$ and $\Delta \text{ADC}$ :

\begin{aligned} \text{AB} &= \text{AD} \quad (\text{Sides of a square are equal}) \\ \text{BC} &= \text{DC} \quad (\text{Sides of a square are equal}) \\ \text{AC} &= \text{AC} \quad (\text{Common side}) \end{aligned}

Therefore, by the $\text{SSS}$ congruence rule: $\Delta \text{ABC} \cong \Delta \text{ADC}$

Answer

(i) $\Delta \text{ABC} \cong \Delta \text{ADC}$ by $\text{SSS}$ congruence.

(ii) Is $\Delta \text{ABC}$ also congruent to $\Delta \text{CDA}$ ?

Step 1 · Check Congruence for $\Delta \text{ABC}$ and $\Delta \text{CDA}$

In $\Delta \text{ABC}$ and $\Delta \text{CDA}$ :

\begin{aligned} \text{AB} &= \text{CD} \quad (\text{Sides of a square are equal}) \\ \text{BC} &= \text{DA} \quad (\text{Sides of a square are equal}) \\ \text{AC} &= \text{CA} \quad (\text{Common diagonal}) \end{aligned}

Therefore, by the $\text{SSS}$ congruence rule: $\Delta \text{ABC} \cong \Delta \text{CDA}$

Answer

(ii) Yes, $\Delta \text{ABC} \cong \Delta \text{CDA}$ .

(iii) Give an example of two triangles where one is congruent to the other in six different ways.

Step 1 · List Congruences for Two Equilateral Triangles

Let $\Delta \text{HEN}$ and $\Delta \text{BIG}$ be two congruent equilateral triangles.

Since all three sides and all three interior angles ( $60^\circ$ ) are equal, every permutation of vertices forms a valid congruence:

$\Delta \text{HEN} \cong \Delta \text{BIG}$
$\Delta \text{HNE} \cong \Delta \text{BGI}$
$\Delta \text{EHN} \cong \Delta \text{IBG}$
$\Delta \text{ENH} \cong \Delta \text{IGB}$
$\Delta \text{NHE} \cong \Delta \text{GBI}$
$\Delta \text{NEH} \cong \Delta \text{GIB}$

Answer

(iii) Two congruent equilateral triangles (e.g., $\Delta \text{HEN} \cong \Delta \text{BIG}$ ) are congruent in $6$ different ways.

Common Mistakes

Order of Vertices: Writing $\Delta \text{ABC} \cong \Delta \text{ADC}$ indicates the vertex mapping $\text{A} \leftrightarrow \text{A}$ , $\text{B} \leftrightarrow \text{D}$ , $\text{C} \leftrightarrow \text{C}$ , whereas $\Delta \text{ABC} \cong \Delta \text{CDA}$ indicates $\text{A} \leftrightarrow \text{C}$ , $\text{B} \leftrightarrow \text{D}$ , $\text{C} \leftrightarrow \text{A}$ . Both are geometrically valid for the square's diagonal.
General Triangles vs. Symmetrical Triangles: Scalene triangles can only be congruent in $1$ way, isosceles in $2$ ways, and equilateral triangles in $6$ ways.

Question 17

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