The problem you shared involves finding the missing sides of right-angled triangles using the Pythagorean Theorem. Let's solve each part step by step.

Problem 1:

a) Find the length of x:

The sides of the triangle are 3.4 cm and 5.1 cm. This is a right-angled triangle, so we apply the Pythagorean theorem:

$a^2 + b^2 = c^2$

where $a = 3.4 \, \text{cm}$, $b = 5.1 \, \text{cm}$, and $c = x$ (the hypotenuse). We need to find $x$.

$3.4^2 + 5.1^2 = x^2$ $11.56 + 26.01 = x^2$ $37.57 = x^2$ $x = \sqrt{37.57} \approx 6.1 \, \text{cm}$

b) Find the length of y:

The triangle has sides of 6 m and 8 m. We need to find the missing side $y$, which is the hypotenuse.

Using the Pythagorean theorem again:

$6^2 + 8^2 = y^2$ $36 + 64 = y^2$ $100 = y^2$ $y = \sqrt{100} = 10 \, \text{m}$

c) Find the length of z:

This is another right-angled triangle with legs 3 mm and 4 mm. We use the Pythagorean theorem to find $z$ (the hypotenuse):

$3^2 + 4^2 = z^2$ $9 + 16 = z^2$ $25 = z^2$ $z = \sqrt{25} = 5 \, \text{mm}$

Problem 2:

a) Find the length of d:

The triangle has sides of 6.4 mm and 10.2 mm. We apply the Pythagorean theorem:

$6.4^2 + 10.2^2 = d^2$ $40.96 + 104.04 = d^2$ $145 = d^2$ $d = \sqrt{145} \approx 12.0 \, \text{mm}$

b) Find the length of e:

The triangle has sides 8 cm and 12 cm. We need to find the missing side $e$, which is the hypotenuse:

$8^2 + 12^2 = e^2$ $64 + 144 = e^2$ $208 = e^2$ $e = \sqrt{208} \approx 14.4 \, \text{cm}$

c) Find the length of f:

This triangle has sides of 11 cm and 14.1 cm. We need to find $f$ (the hypotenuse):

$11^2 + 14.1^2 = f^2$ $121 + 198.81 = f^2$ $319.81 = f^2$ $f = \sqrt{319.81} \approx 17.9 \, \text{cm}$

Summary of Answers:

1. a) $x \approx 6.1 \, \text{cm}$ b) $y = 10 \, \text{m}$ c) $z = 5 \, \text{mm}$

2. a) $d \approx 12.0 \, \text{mm}$ b) $e \approx 14.4 \, \text{cm}$ c) $f \approx 17.9 \, \text{cm}$

Let me know if you'd like any further explanations or help with additional questions!

Here are a few more related questions to deepen the topic:

1. How do you use the Pythagorean theorem when you only know one leg of the triangle and the hypotenuse?
2. What happens if the triangle is not a right-angled triangle? How do you find the missing sides?
3. Can the Pythagorean theorem be used in non-Euclidean geometry?
4. What is the distance formula and how does it relate to the Pythagorean theorem?
5. How do you apply the Pythagorean theorem to find the diagonal of a rectangle?

Tip: The Pythagorean theorem only works for right-angled triangles, so always verify that the triangle is a right-angled one before applying it!