Question

period: 4name: angel carventeb - 9.2 pythagorean theorem and special cases -lab worksheetin exercises 1-4, find the value of x. write your answer in simplest form.1.2.3.4.in exercises 5-7, find the values of x and y. write your answers in simplest form.5.6.7.

Explanation:

Exercise 1

Step1: Identify triangle type

It is a 45-45-90 right triangle, so legs are equal: the unknown leg is 10.

Step2: Calculate hypotenuse $x$

Use 45-45-90 triangle rule: hypotenuse = leg $\times \sqrt{2}$
$x = 10\sqrt{2}$

Exercise 2

Step1: Identify triangle type

It is a 45-45-90 right triangle with legs $\sqrt{2}$.

Step2: Calculate hypotenuse $x$

Use 45-45-90 triangle rule: hypotenuse = leg $\times \sqrt{2}$
$x = \sqrt{2} \times \sqrt{2} = 2$

Exercise 3

Step1: Identify triangle type

It is a 45-45-90 right triangle with hypotenuse $8\sqrt{2}$.

Step2: Calculate leg $x$

Use 45-45-90 triangle rule: leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$
$x = \frac{8\sqrt{2}}{\sqrt{2}} = 8$

Exercise 4

Step1: Identify triangle type

It is a 45-45-90 right triangle with hypotenuse 12.

Step2: Calculate leg $x$

Use 45-45-90 triangle rule: leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$, rationalize denominator
$x = \frac{12}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}$

Exercise 5

Step1: Identify triangle type

It is a 30-60-90 right triangle, opposite side to 30° is 15.

Step2: Calculate hypotenuse $y$

Use 30-60-90 rule: hypotenuse = 2 × shorter leg
$y = 2 \times 15 = 30$

Step3: Calculate longer leg $x$

Use 30-60-90 rule: longer leg = shorter leg $\times \sqrt{3}$
$x = 15\sqrt{3}$

Exercise 6

Step1: Identify triangle type

It is a 30-60-90 right triangle, hypotenuse is 22.

Step2: Calculate shorter leg $x$

Use 30-60-90 rule: shorter leg = $\frac{\text{hypotenuse}}{2}$
$x = \frac{22}{2} = 11$

Step3: Calculate longer leg $y$

Use 30-60-90 rule: longer leg = shorter leg $\times \sqrt{3}$
$y = 11\sqrt{3}$

Exercise 7

Step1: Identify triangle type

It is a 30-60-90 right triangle, longer leg is 9.

Step2: Calculate shorter leg $x$

Use 30-60-90 rule: shorter leg = $\frac{\text{longer leg}}{\sqrt{3}}$, rationalize denominator
$x = \frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}$

Step3: Calculate hypotenuse $y$

Use 30-60-90 rule: hypotenuse = 2 × shorter leg
$y = 2 \times 3\sqrt{3} = 6\sqrt{3}$

Answer:

1. $x = 10\sqrt{2}$
2. $x = 2$
3. $x = 8$
4. $x = 6\sqrt{2}$
5. $x = 15\sqrt{3}$, $y = 30$
6. $x = 11$, $y = 11\sqrt{3}$
7. $x = 3\sqrt{3}$, $y = 6\sqrt{3}$