Rational Numbers | FIO

Question 8

In the following pattern, fill in the missing numbers:

\begin{aligned} 1^2 + 2^2 + 2^2 &= 3^2 \\ 2^2 + 3^2 + 6^2 &= 7^2 \\ 3^2 + 4^2 + 12^2 &= 13^2 \end{aligned}

(a) $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$

(b) $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$

Solution

Understand the Question

The pattern follows the general form $a^2 + b^2 + c^2 = d^2$ where:
- The third base is the product of the first two bases: $c = a \times b$
- The base on the right-hand side is one more than the third base: $d = c + 1$
Verification from given examples:
- Row 1: $1^2 + 2^2 + 2^2 = 3^2 \implies c = 1 \times 2 = 2$ and $d = 2 + 1 = 3$
- Row 2: $2^2 + 3^2 + 6^2 = 7^2 \implies c = 2 \times 3 = 6$ and $d = 6 + 1 = 7$
- Row 3: $3^2 + 4^2 + 12^2 = 13^2 \implies c = 3 \times 4 = 12$ and $d = 12 + 1 = 13$

(a) $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$

Step 1 · Find the Missing Number

Given $a = 4$ , $b = 5$ , and $c = 20$ .

Verify the third number:

\begin{aligned} a \times b &= 4 \times 5 \\ &= 20 \end{aligned}

Find the number on the right-hand side ( $d$ ):

\begin{aligned} d &= c + 1 \\ &= 20 + 1 \\ &= 21 \end{aligned}

Thus, the equation is: $4^2 + 5^2 + 20^2 = (21)^2$

Answer

(a) $21$

(b) $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$

Step 1 · Find the Missing Numbers

Given $a = 9$ and $b = 10$ .

Find the third number ( $c$ ):

\begin{aligned} c &= a \times b \\ &= 9 \times 10 \\ &= 90 \end{aligned}

Find the number on the right-hand side ( $d$ ):

\begin{aligned} d &= c + 1 \\ &= 90 + 1 \\ &= 91 \end{aligned}

Thus, the equation is: $9^2 + 10^2 + (90)^2 = (91)^2$

Answer

(b) $90$ and $91$

Common Mistakes

Adding Instead of Multiplying: Adding the first two base numbers ( $a + b$ ) instead of multiplying them ( $a \times b$ ) to find the third base.
Manual Computation: Trying to compute large squares directly instead of using the pattern relation $d = c + 1$ .

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