MBA Nexus - CAT & OMETS 2024, [18-10-2024 10:09]
Q7 If ab : bc : ac = 3 : 4 : 5, then find a²: b² : c² .

MBA Nexus - CAT & OMETS 2024, [18-10-2024 10:10]
[ Poll : Ans ]
- 9 : 16 : 25
- 81 : 256 : 625
- 225 : 144 : 400
- 144 : 225 : 400

We are given the ratio \( ab : bc : ac = 3 : 4 : 5 \), and we need to find the ratio of \( a^2 : b^2 : c^2 \).

### Step 1: Express the ratios of the products of sides

The given ratio implies the following relationships between the products of sides:
\[
ab : bc : ac = 3 : 4 : 5
\]

We can set the values of these products as:
\[
ab = 3k, \quad bc = 4k, \quad ac = 5k
\]
where \( k \) is some constant.

### Step 2: Solve for the individual variables \( a \), \( b \), and \( c \)

Now, divide the products to express \( a \), \( b \), and \( c \) in terms of each other:

1. From \( \frac{ab}{ac} = \frac{3}{5} \), we get:
   \[
   \frac{b}{c} = \frac{3}{5} \quad \text{or} \quad b = \frac{3}{5}c
   \]

2. From \( \frac{bc}{ac} = \frac{4}{5} \), we get:
   \[
   \frac{b}{a} = \frac{4}{5} \quad \text{or} \quad b = \frac{4}{5}a
   \]

### Step 3: Substitute and calculate

Using the relation \( b = \frac{4}{5}a \) in the equation \( b = \frac{3}{5}c \), we get:
\[
\frac{4}{5}a = \frac{3}{5}c
\]
This simplifies to:
\[
4a = 3c \quad \text{or} \quad c = \frac{4}{3}a
\]

Now we have:
- \( b = \frac{4}{5}a \)
- \( c = \frac{4}{3}a \)

### Step 4: Find the squares

Now, let's find the squares of \( a \), \( b \), and \( c \) in terms of \( a^2 \):

1. \( b^2 = \left( \frac{4}{5} a \right)^2 = \frac{16}{25} a^2 \)
2. \( c^2 = \left( \frac{4}{3} a \right)^2 = \frac{16}{9} a^2 \)

### Step 5: Find the ratio \( a^2 : b^2 : c^2 \)

The ratio becomes:
\[
a^2 : b^2 : c^2 = a^2 : \frac{16}{25} a^2 : \frac{16}{9} a^2
\]

To simplify, multiply through by the least common denominator (LCD) of 225:
\[
225a^2 : 144a^2 : 400a^2
\]

Thus, the ratio is:
\[
225 : 144 : 400
\]

### Final Answer: 
The correct ratio is \( 225 : 144 : 400 \), which corresponds to the third option.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How can we generalize ratios involving more than three variables?
2. How does the ratio of squares change when we manipulate products like \( ab, bc, \) and \( ac \)?
3. What is the geometric significance of the ratio \( ab : bc : ac \)?
4. How can we apply this method to find the cubes of the variables?
5. How does the concept of proportions help in solving similar triangle problems?

**Tip:** When solving ratios, simplify expressions step by step to avoid errors in complex relationships.

If ab : bc : ac = 3 : 4 : 5, then find a²: b² : c².

Learn how to solve the ratio problem where ab : bc : ac = 3 : 4 : 5 and find a² : b² : c². This solution covers ratio manipulation, algebraic relationships, and key mathematical concepts. Suitable for high school students.

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