Let's address the problems shown in the image step by step.

1. Work out the value of $a$. Justify your answer.

The triangle has angles $113^\circ, 48^\circ,$ and $a$. The sum of the angles in a triangle is always $180^\circ$.

$a = 180^\circ - 113^\circ - 48^\circ = 65^\circ$

So, $a = 65^\circ$.

2. Work out the size of the largest angle in a pentagon.

The angles are given as $x^\circ, x^\circ, x^\circ, x^\circ,$ and $(x+10)^\circ$. The sum of the interior angles of a pentagon is:

$(5 - 2) \times 180^\circ = 540^\circ$

Setting up the equation for the angles:

$4x + (x + 10) = 540$

$5x + 10 = 540$

$5x = 530$

$x = 106^\circ$

The largest angle is:

$x + 10 = 106^\circ + 10^\circ = 116^\circ$

3. How many sides does the polygon have?

Three of the interior angles are given as $AB = BC = CD = 36^\circ$. For a regular polygon, each exterior angle is:

$\text{Exterior Angle} = \frac{360^\circ}{n}$

Here, $\text{Exterior Angle} = 36^\circ$, so:

$n = \frac{360^\circ}{36^\circ} = 10$

The polygon has 10 sides.

4a. Construct a square inside a circle with radius 6 cm.

You can construct the square using a ruler and compass. Follow these steps:

1. Draw a circle with radius 6 cm.
2. Mark the center of the circle as $O$.
3. Draw two perpendicular diameters of the circle to create 4 points where the square’s vertices will lie.
4. Connect these points to form a square.

4b. Measure the side of the square.

The diagonal of the square is equal to the diameter of the circle, which is $12 \, \text{cm}$ (twice the radius). Using the Pythagorean theorem:

$\text{Diagonal} = \sqrt{2 \times (\text{Side})^2}$

$12 = \sqrt{2} \cdot \text{Side}$

$\text{Side} = \frac{12}{\sqrt{2}} = 6\sqrt{2} \, \text{cm} \approx 8.49 \, \text{cm}$

5. How long is the diagonal of the square with side 25 m?

Using the Pythagorean theorem for a square’s diagonal:

$\text{Diagonal} = \sqrt{2 \times (\text{Side})^2}$

$\text{Diagonal} = \sqrt{2 \times 25^2} = \sqrt{1250} \approx 35.36 \, \text{m}$

6. Calculate the lengths $x$ and $y$ in the right triangle.

Given:

• The hypotenuse = $26$,
• One leg = $8$,
• Another leg = $6$.

Using the Pythagorean theorem:

$x^2 + y^2 = 26^2$

$8^2 + y^2 = 26^2$

$64 + y^2 = 676$

$y^2 = 676 - 64 = 612$

$y = \sqrt{612} \approx 24.74$

For $x$:

$x^2 + 6^2 = 26^2$

$x^2 + 36 = 676$

$x^2 = 640$

$x = \sqrt{640} \approx 25.30$

Would you like further details or additional explanations? Here are some questions to expand your understanding:

1. What is the formula for calculating the sum of interior angles of a polygon?
2. How do you determine the largest angle in an irregular polygon?
3. Can you derive the formula for the side length of a square from its diagonal?
4. How can you use a compass to construct other regular polygons?
5. What are the applications of the Pythagorean theorem in real-world geometry problems?

Tip: For constructing polygons inside circles, the circle's radius determines the vertices' positions, which can be marked using equal angular divisions.