Finding Side Length in Congruent Right Triangles ABC and PQR

We are given two triangles, ABC and PQR, which are congruent. This means that their corresponding sides and angles are equal in measure. The congruence statement is ABC $\cong$ PQR.

We are also given that both triangles are right-angled triangles, with the right angles at vertices A and P, respectively. So, $\angle A = 90^\circ$ and $\angle P = 90^\circ$.

The given side lengths are BC = 13 cm and PR = 5 cm.

Understanding Congruent Triangles Properties

When two triangles are congruent, their corresponding parts (sides and angles) are equal. From the congruence statement ABC $\cong$ PQR, we can identify the corresponding sides:

• AB corresponds to PQ, so AB = PQ.
• BC corresponds to QR, so BC = QR.
• AC corresponds to PR, so AC = PR.

Similarly, the corresponding angles are:

• $\angle A$ corresponds to $\angle P$, so $\angle A = \angle P$. (Given as $90^\circ$)
• $\angle B$ corresponds to $\angle Q$, so $\angle B = \angle Q$.
• $\angle C$ corresponds to $\angle R$, so $\angle C = \angle R$.

Applying Given Information to Triangle ABC

We are given BC = 13 cm. Since BC corresponds to QR, we know QR = 13 cm.

We are given PR = 5 cm. Since AC corresponds to PR, we know AC = 5 cm.

We are looking for the length of side AB in triangle ABC. We now know the following about triangle ABC:

• $\angle A = 90^\circ$
• BC = 13 cm (Hypotenuse, as it is opposite the right angle)
• AC = 5 cm (One leg)
• AB = ? (The other leg)

Using the Pythagorean Theorem

Since triangle ABC is a right-angled triangle at A, we can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (the legs).

For triangle ABC, the Pythagorean theorem is:

\( (Hypotenuse)^2 = (Leg\ 1)^2 + (Leg\ 2)^2 \)

\( BC^2 = AB^2 + AC^2 \)

Calculating the Length of AB

Substitute the known values into the Pythagorean theorem equation:

\( 13^2 = AB^2 + 5^2 \)

Calculate the squares:

\( 169 = AB^2 + 25 \)

To find \( AB^2 \), subtract 25 from 169:

\( AB^2 = 169 - 25 \)

\( AB^2 = 144 \)

To find AB, take the square root of 144:

\( AB = \sqrt{144} \)

\( AB = 12 \)

Since AB is a length, it must be a positive value. Therefore, AB = 12 cm.

We can verify this using the Pythagorean triple (5, 12, 13), which corresponds to the sides of a right-angled triangle.

Conclusion

Using the properties of congruent triangles and the Pythagorean theorem, we found that the length of side AB is 12 cm.

Revision Table: Key Concepts

Additional Information: Congruence Criteria

Besides knowing that corresponding parts of congruent triangles are equal (CPCTC), there are criteria to determine if two triangles are congruent without checking all sides and angles. Some common criteria are:

• SSS (Side-Side-Side): If three sides of one triangle are equal to three corresponding sides of another triangle.
• SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two corresponding sides and the included angle of another triangle.
• ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two corresponding angles and the included side of another triangle.
• AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two corresponding angles and the corresponding non-included side of another triangle.
• RHS (Right-angle-Hypotenuse-Side): For right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of another right-angled triangle.

In this specific problem, knowing $\triangle ABC \cong \triangle PQR$ is given, so we directly used the property of corresponding sides being equal (AC = PR). We then used the Pythagorean theorem as $\triangle ABC$ is a right-angled triangle with two sides known.