A) In this part, we are given an independent variable – an acute angle V in a right triangle, dependent variable - area B of the triangle and a constant – hypotenuse G has a fixed length.

We can first express the two legs of the right triangle in terms of hypotenuse G and acute angle V as shown below:

Height \(\displaystyle={G}{\sin{{\left({V}\right)}}}\)

Base \(\displaystyle={G}{\cos{{\left({V}\right)}}}\)

Use the formula for area of triangle as shown below:

Area \(\displaystyle=\frac{{1}}{{2}}\) *Base*Height

Area \(\displaystyle=\frac{{1}}{{2}}\cdot{\left({G}{\cos{{\left({V}\right)}}}\right)}\cdot{\left({G}{\sin{{\left({V}\right)}}}\right)}\)

Area \(\displaystyle=\frac{{1}}{{2}}{G}^{{2}}{\sin{{\left({V}\right)}}}{\cos{{\left({V}\right)}}}\)

Area \(\displaystyle=\frac{{1}}{{4}}{G}^{{2}}{\sin{{\left({2}{V}\right)}}}\)

Domain of this function is \(\displaystyle{\left({0},\frac{\pi}{{2}}\right)}\) and range is \(\displaystyle{\left({0},\frac{{G}^{{2}}}{{4}}\right)}\).

Since area function is oscillating function, therefore, the relation between two variables is neither positive and nor negative.

B) In this part, we are given an independent variable – a leg P of a right triangle, dependent variable - the hypotenuse G of the right triangle, and constant – second leg of length 2.

We can use Pythagorean theorem to express the relationship between P, G and 2 as shown below:

\(\displaystyle{P}^{{2}}+{2}^{{2}}\)

\(\displaystyle{G}^{{2}}={P}^{{2}}+{4}\)

\(\displaystyle{G}=\sqrt{{{P}^{{2}}+{4}}}\)

In reference to the given question, since P represents a leg of a right triangle, it can take any real number greater than 0. Therefore, domain of this function is \(\displaystyle{\left({0},\infty\right)}\).

Range of this function is all real numbers greater than 2, that is, \(\displaystyle{\left({2},\infty\right)}\).

Since value of hypotenuse increases as the length of leg increases, therefore, there is a positive relationship between the two variables.

C) In this part, we are given an independent variable – the hypotenuse G of a right triangle, dependent variable - the leg P of the right triangle, and constant – second leg of length 5.

We can use Pythagorean theorem to express the relationship between P, G and 2 as shown below:

\(\displaystyle{P}^{{2}}+{5}^{{2}}={G}^{{2}}\)

\(\displaystyle{P}^{{2}}={G}^{{2}}-{25}\)

\(\displaystyle{P}=\sqrt{{{G}^{{2}}-{25}}}\)

G can take any real number greater than 5 in order for this function to exist. Therefore, domain of this function is \(\displaystyle{\left({5},\infty\right)}\).

Range of this function is all positive real numbers, that is, \(\displaystyle{\left({0},\infty\right)}\).

Since value of leg increases as the length of hypotenuse increases, therefore, there is a positive relationship between the two variables.

We can first express the two legs of the right triangle in terms of hypotenuse G and acute angle V as shown below:

Height \(\displaystyle={G}{\sin{{\left({V}\right)}}}\)

Base \(\displaystyle={G}{\cos{{\left({V}\right)}}}\)

Use the formula for area of triangle as shown below:

Area \(\displaystyle=\frac{{1}}{{2}}\) *Base*Height

Area \(\displaystyle=\frac{{1}}{{2}}\cdot{\left({G}{\cos{{\left({V}\right)}}}\right)}\cdot{\left({G}{\sin{{\left({V}\right)}}}\right)}\)

Area \(\displaystyle=\frac{{1}}{{2}}{G}^{{2}}{\sin{{\left({V}\right)}}}{\cos{{\left({V}\right)}}}\)

Area \(\displaystyle=\frac{{1}}{{4}}{G}^{{2}}{\sin{{\left({2}{V}\right)}}}\)

Domain of this function is \(\displaystyle{\left({0},\frac{\pi}{{2}}\right)}\) and range is \(\displaystyle{\left({0},\frac{{G}^{{2}}}{{4}}\right)}\).

Since area function is oscillating function, therefore, the relation between two variables is neither positive and nor negative.

B) In this part, we are given an independent variable – a leg P of a right triangle, dependent variable - the hypotenuse G of the right triangle, and constant – second leg of length 2.

We can use Pythagorean theorem to express the relationship between P, G and 2 as shown below:

\(\displaystyle{P}^{{2}}+{2}^{{2}}\)

\(\displaystyle{G}^{{2}}={P}^{{2}}+{4}\)

\(\displaystyle{G}=\sqrt{{{P}^{{2}}+{4}}}\)

In reference to the given question, since P represents a leg of a right triangle, it can take any real number greater than 0. Therefore, domain of this function is \(\displaystyle{\left({0},\infty\right)}\).

Range of this function is all real numbers greater than 2, that is, \(\displaystyle{\left({2},\infty\right)}\).

Since value of hypotenuse increases as the length of leg increases, therefore, there is a positive relationship between the two variables.

C) In this part, we are given an independent variable – the hypotenuse G of a right triangle, dependent variable - the leg P of the right triangle, and constant – second leg of length 5.

We can use Pythagorean theorem to express the relationship between P, G and 2 as shown below:

\(\displaystyle{P}^{{2}}+{5}^{{2}}={G}^{{2}}\)

\(\displaystyle{P}^{{2}}={G}^{{2}}-{25}\)

\(\displaystyle{P}=\sqrt{{{G}^{{2}}-{25}}}\)

G can take any real number greater than 5 in order for this function to exist. Therefore, domain of this function is \(\displaystyle{\left({5},\infty\right)}\).

Range of this function is all positive real numbers, that is, \(\displaystyle{\left({0},\infty\right)}\).

Since value of leg increases as the length of hypotenuse increases, therefore, there is a positive relationship between the two variables.