Let's go through the problem step-by-step.

Problem Summary

The number of diagonals $D$ in a polygon is given by: $D = Xn^2 + Yn$ where $n$ is the number of sides.

We are asked to:

1. Use the number of diagonals in a triangle and a quadrilateral to show that the equations $3X + Y = 0$ and $8X + 2Y = 1$ hold.
2. Solve these simultaneous equations for $X$ and $Y$.
3. Use the values of $X$ and $Y$ to find the number of diagonals in an octagon.

Solution

(a) Formulating the Equations

For a polygon with $n$ sides:

• Triangle ($n = 3$): A triangle has 0 diagonals. $D = X(3)^2 + Y(3) = 9X + 3Y$ Since $D = 0$, we get: $9X + 3Y = 0$ Dividing by 3, we obtain: $3X + Y = 0$

• Quadrilateral ($n = 4$): A quadrilateral has 2 diagonals. $D = X(4)^2 + Y(4) = 16X + 4Y$ Since $D = 2$, we get: $16X + 4Y = 2$ Dividing by 2, we obtain: $8X + 2Y = 1$

These equations match the ones given in the problem: $3X + Y = 0 \quad \text{and} \quad 8X + 2Y = 1$

(b) Solving for $X$ and $Y$

We have the system:

1. $3X + Y = 0$
2. $8X + 2Y = 1$

Let's solve these equations simultaneously.

From the first equation: $Y = -3X$

Substitute $Y = -3X$ into the second equation: $8X + 2(-3X) = 1$ $8X - 6X = 1$ $2X = 1$ $X = \frac{1}{2}$

Now, substitute $X = \frac{1}{2}$ back into $Y = -3X$: $Y = -3\left(\frac{1}{2}\right) = -\frac{3}{2}$

So, the values are: $X = \frac{1}{2}, \quad Y = -\frac{3}{2}$

(c) Finding the Number of Diagonals in an Octagon ($n = 8$)

For an octagon ($n = 8$): $D = Xn^2 + Yn$ Substitute $X = \frac{1}{2}$, $Y = -\frac{3}{2}$, and $n = 8$: $D = \frac{1}{2}(8)^2 + \left(-\frac{3}{2}\right)(8)$ $D = \frac{1}{2}(64) - \frac{3}{2}(8)$ $D = 32 - 12 = 20$

So, the number of diagonals in an octagon is 20.

Final Answers

• (a) We verified the equations are correct.
• (b) $X = \frac{1}{2}$ and $Y = -\frac{3}{2}$.
• (c) The number of diagonals in an octagon is 20.

Would you like further clarification on any step?

Follow-up Questions:

1. How can we derive the general formula for the number of diagonals in a polygon?
2. What would happen if we used different polygons to verify the equations?
3. How do these equations relate to the polygon diagonal formula $D = \frac{n(n-3)}{2}$?
4. How would the formula change if we considered 3D shapes instead of 2D polygons?
5. Can we generalize this method for polygons with an arbitrary number of diagonals?

Tip:

When solving simultaneous equations, always check if one equation can be simplified to express one variable in terms of the other. This simplifies substitution and reduces calculation steps.