The problem asks us to find the missing elements (sides and angles), the perimeter, and the area of the given right triangles. Triangle a) has side $a=45m$ and angle $C=25^\circ$ given. Triangle b) has sides $5km$ and $12km$ given.

1. Problem Description

The problem asks us to find the missing elements (sides and angles), the perimeter, and the area of the given right triangles.

Triangle a) has side $a=45m$ and angle $C=25^\circ$ given.

Triangle b) has sides $5km$ and $12km$ given.

2. Solution Steps

a) For triangle a):

We have a right triangle with side $a = 45 m$ and angle $C = 25^\circ$.

Since it's a right triangle, angle $A = 90^\circ$.

The sum of angles in a triangle is $180^\circ$, so $B = 180^\circ - 90^\circ - 25^\circ = 65^\circ$.

Now, we can find side $b$ using trigonometric functions:

$\sin(C) = \frac{b}{a}$

$b = a \sin(C) = 45 \sin(25^\circ) \approx 45 \times 0.4226 = 19.017 m$

We can find side $c$ using the Pythagorean theorem or trigonometric functions:

$\cos(C) = \frac{c}{a}$

$c = a \cos(C) = 45 \cos(25^\circ) \approx 45 \times 0.9063 = 40.7835 m$

The perimeter $P = a + b + c = 45 + 19.017 + 40.7835 = 104.8005 m$

The area $Area = \frac{1}{2} b c = \frac{1}{2} \times 19.017 \times 40.7835 \approx 387.60 m^2$

b) For triangle b):

We have a right triangle with legs $5km$ and $12km$. Let $a=5km$ and $b=12km$.

We can find side $x$ (the hypotenuse) using the Pythagorean theorem:

$x^2 = a^2 + b^2$

$x^2 = 5^2 + 12^2 = 25 + 144 = 169$

$x = \sqrt{169} = 13 km$

Let $\alpha$ be the angle opposite to side $a$, so $\tan(\alpha) = \frac{a}{b} = \frac{5}{12}$.

$\alpha = \arctan(\frac{5}{12}) \approx 22.62^\circ$

Let $\beta$ be the angle opposite to side $b$, so $\tan(\beta) = \frac{b}{a} = \frac{12}{5}$.

$\beta = \arctan(\frac{12}{5}) \approx 67.38^\circ$

The perimeter $P = a + b + x = 5 + 12 + 13 = 30 km$

The area $Area = \frac{1}{2} a b = \frac{1}{2} \times 5 \times 12 = 30 km^2$

3. Final Answer

a) For triangle a):

$B = 65^\circ$

$b \approx 19.017 m$

$c \approx 40.7835 m$

$P \approx 104.8005 m$

$Area \approx 387.60 m^2$

b) For triangle b):

$x = 13 km$

$\alpha \approx 22.62^\circ$

$\beta \approx 67.38^\circ$

$P = 30 km$

$Area = 30 km^2$