Question

1. a right triangle with a 45° angle, hypotenuse 2, legs x and y. 6) a right triangle with a 45° angle, one leg (5sqrt{2}), hypotenuse a, leg b. 7) a right triangle with a 45° angle, hypotenuse 10, legs z and v. 8) a right triangle with a 45° angle, one leg (\frac{9sqrt{2}}{2}), hypotenuse u, leg v. 9) a right triangle with a 45° angle, hypotenuse (3sqrt{5}), legs y and x. 10) a right triangle with a 45° angle, one leg 5, legs y and x, hypotenuse... (description of each triangle with 45° angle and right angle, sides labeled with variables or values)

Explanation:

Problem 5

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $x=y$.

Step2: Use Pythagorean theorem

$$x^2 + y^2 = 2^2$$
Substitute $x=y$:
$$2x^2 = 4$$
$$x^2=2$$
$$x=y=\sqrt{2}$$

Problem 6

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $b=5\sqrt{2}$.

Step2: Calculate hypotenuse $a$

$$a = b\sqrt{2} = 5\sqrt{2} \times \sqrt{2} = 5\times2=10$$

Problem 7

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $w=v$.

Step2: Use Pythagorean theorem

$$w^2 + v^2 = 10^2$$
Substitute $w=v$:
$$2w^2 = 100$$
$$w^2=50$$
$$w=v=5\sqrt{2}$$

Problem 8

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $v=\frac{9\sqrt{2}}{2}$.

Step2: Calculate hypotenuse $u$

$$u = v\sqrt{2} = \frac{9\sqrt{2}}{2} \times \sqrt{2} = \frac{9\times2}{2}=9$$

Problem 9

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $x=y$.

Step2: Use Pythagorean theorem

$$x^2 + y^2 = (3\sqrt{5})^2$$
Substitute $x=y$:
$$2x^2 = 9\times5=45$$
$$x^2=\frac{45}{2}$$
$$x=y=\frac{3\sqrt{10}}{2}$$

Problem 10

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $y=5$.

Step2: Calculate hypotenuse $x$

$$x = y\sqrt{2} = 5\sqrt{2}$$

Answer:

1. $x=\sqrt{2}$, $y=\sqrt{2}$
2. $a=10$, $b=5\sqrt{2}$
3. $w=5\sqrt{2}$, $v=5\sqrt{2}$
4. $u=9$, $v=\frac{9\sqrt{2}}{2}$
5. $x=\frac{3\sqrt{10}}{2}$, $y=\frac{3\sqrt{10}}{2}$
6. $x=5\sqrt{2}$, $y=5$