In a right angled triangle, the hypotenuse is 20 cm and the other two sides are in the ratio 4 : 3. What are the lengths (in cm) of these two legs of the triangle?

Introduction / Context:
This problem is a standard application of the Pythagoras theorem in a right angled triangle when you are given the hypotenuse and the ratio of the remaining two sides. Such questions are very common in aptitude and competitive exams and help you practice working with integer Pythagorean triples.

Given Data / Assumptions:

• The triangle is right angled.
• The hypotenuse has length 20 cm.
• The other two sides (legs) are in the ratio 4 : 3.
• Let the legs be 4k cm and 3k cm for some positive real number k.
• We use the Pythagoras theorem: (leg1)^2 + (leg2)^2 = (hypotenuse)^2.

Concept / Approach:
Whenever the ratio of the legs is given, a very efficient way is to express the unknown sides in terms of a common factor k. Then, apply the Pythagoras theorem and solve for k. Once k is known, you immediately get the exact lengths of the legs. Here, the ratio 4 : 3 hints at a scaled version of the classic 3 4 5 Pythagorean triple.

Step-by-Step Solution:
Step 1: Let the two legs be 4k cm and 3k cm. Step 2: Apply Pythagoras theorem: (4k)^2 + (3k)^2 = 20^2. Step 3: Compute the squares: 16k^2 + 9k^2 = 25k^2. Step 4: Set 25k^2 = 400 (since 20^2 = 400). Step 5: Divide both sides by 25: k^2 = 400 / 25 = 16. Step 6: Take the positive square root (lengths are positive): k = 4. Step 7: Therefore, the legs are 4k = 4 * 4 = 16 cm and 3k = 3 * 4 = 12 cm.

Verification / Alternative check:
Check the Pythagorean relation with the found lengths: 16^2 + 12^2 = 256 + 144 = 400, and the hypotenuse squared is 20^2 = 400. The equality holds exactly, so 16 cm and 12 cm are consistent. Also, the ratio 16 : 12 simplifies to 4 : 3, matching the given ratio of the legs.

Why Other Options Are Wrong:
4 cm and 3 cm, 8 cm and 6 cm, and 12 cm and 9 cm all keep the 4 : 3 ratio but do not satisfy the Pythagoras theorem with hypotenuse 20 cm. For each of these, the sum of squares of the legs is not 400. The pair 10 cm and 8 cm does not even preserve the 4 : 3 ratio and also fails the Pythagorean condition for hypotenuse 20 cm.

Common Pitfalls:
A frequent mistake is to scale a known triple incorrectly or to forget to square the scaling factor when checking the Pythagoras relation. Another error is to choose numbers in the correct ratio but not check whether they fit the given hypotenuse. Always use (leg1)^2 + (leg2)^2 = hypotenuse^2 to confirm the final pair of sides.

Final Answer:
The two legs of the triangle are 16 cm and 12 cm.