The three triangles A, B, and C are shown below - Junior Cycle Mathematics - Question 4 - 2022

The perimeter of Triangle B can be calculated using the formula:

$P_B = 3 + (2x + 1) + (2x + 1)$

Combining the terms, we have:

$P_B = 3 + 2x + 1 + 2x + 1 = 4 + 4x \text{ cm}.$

To find the value of $x$, we set the perimeter equal to 24:

$4 + 4x = 24$

Subtracting 4 from both sides yields:

$4x = 24 - 4$

So:

$4x = 20$

Dividing both sides by 4 gives:

$x = 5$.

To find the perimeter of Triangle C, substitute $y$ with the previous results and calculate:

$P_C = 5 + y^{2} + 3$

Using the value of $y$ from previous calculations, we know $y = 4$ from part (ii):

$P_C = 5 + (4)^{2} + 3 = 5 + 16 + 3 = 24.$

From the perimeter we calculated: $P_C = 5 + y^{2} + 3$. Setting this equal to $k = 40$ (the next term in linear sequence):

$40 = 5 + y^{2} + 3$

This simplifies to:

$y^{2} = 40 - 8 = 32$

Taking the square root gives:

$y = 4.$

Thus, the value of $y$ is $4$.