Question 8

We will find the rule that connects the numbers in the given pattern and then use this rule to fill in the missing numbers.

Step 1 — Discovering the pattern

Let us look at the numbers in each equation. We can write the general form of the equations as $a^2 + b^2 + c^2 = d^2$.

For the first equation: $1^2 + 2^2 + 2^2 = 3^2$ Here, $a = 1$, $b = 2$, $c = 2$, and $d = 3$. We notice that the third number, $c$, is the product of the first two numbers, $a$ and $b$. $a \times b = 1 \times 2 = 2$ This matches $c$. We also notice that the number on the right side, $d$, is one more than the third number, $c$. $c + 1 = 2 + 1 = 3$ This matches $d$.

For the second equation: $2^2 + 3^2 + 6^2 = 7^2$ Here, $a = 2$, $b = 3$, $c = 6$, and $d = 7$. Let us check if our rule holds. $a \times b = 2 \times 3 = 6$ This matches $c$. $c + 1 = 6 + 1 = 7$ This matches $d$. The pattern works for the second equation too.

For the third equation: $3^2 + 4^2 + 12^2 = 13^2$ Here, $a = 3$, $b = 4$, $c = 12$, and $d = 13$. Let us check our rule again. $a \times b = 3 \times 4 = 12$ This matches $c$. $c + 1 = 12 + 1 = 13$ This matches $d$. The pattern is consistent for all given equations.

So, the pattern is: the third number in the sum is the product of the first two numbers, and the number on the right side is one more than the third number.

Step 2 — Solving part (a)

The equation for part (a) is $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$. Here, the first number $a = 4$. The second number $b = 5$. The third number $c = 20$. Let us verify the third number using our pattern. $a \times b = 4 \times 5$ $= 20$ This matches the given $20^2$. Now, we need to find the number on the right side, which we called $d$. According to our pattern, $d = c + 1$. $d = 20 + 1$ $= 21$ So, the missing number is 21.

$\boxed{4^2 + 5^2 + 20^2 = (21)^2}$

Step 3 — Solving part (b)

The equation for part (b) is $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$. Here, the first number $a = 9$. The second number $b = 10$. First, we find the third number, $c$. According to our pattern, $c = a \times b$. $c = 9 \times 10$ $= 90$ So, the first missing number is 90. Next, we find the number on the right side, $d$. According to our pattern, $d = c + 1$. $d = 90 + 1$ $= 91$ So, the second missing number is 91.

$\boxed{9^2 + 10^2 + (90)^2 = (91)^2}$

Answer

(a) $4^2 + 5^2 + 20^2 = (\mathbf{21})^2$ (b) $9^2 + 10^2 + (\mathbf{90})^2 = (\mathbf{91})^2$