Pythagoream Theorem and Special Right Triangles

Example 1:Q: What is the hypotenuse of the figure below? Show your work and explain how you got your answer.A:  Hypotenuse = 5√2 feet

Steps: Check if there are two angles that are equal to each other (45°.) If so, then you know that the sides the two angles don't share are equal. Now, since you know that two sides are equal, you can use the Pythagorean Theorem to solve the rest (a² + b² = c².) Plug-in the two sides you know into the equation. Solve.

          ↑        5 ft

Things to remember: All triangles' angles are equal to (add up to) 180°. If two angles are equal, the two sides they don't share are equal as well. Even if you only know that one angle of a right triangle is 45°, you know the other one is the same because 90° + 45° + 45° = 180°. If you see something like m that means the measurement of angle D is equal to 106°. If one angle is 60° and another is 30° in a right triangle, then to find the hypotenuse, you must use this formula:  The sign "≅" means congruent in geometry.

Work: 90° + 45° + 45° = 180° m  45° ≅ m Line AC and Line CB = 5 feet a² + b² = c² ... 5² + 5² = c² 25 + 25 = c² 50 = c² c = √50 c = 5√2 5√2 = √50

Explanation:The two angles, A and B, are equal. Since these two angles are congruent, you know that lines AC and CB are congruent as well. Now you know that "a" in the equation is 5 feet as well as "b". 5 squared is 25 times two is 50. So the answer would be √50, but it needs to be simplified more. 25 * 2 = 50 so you would write it as 5√2. Why? Because the square root of 25 is 5 so you put it at the beginning. And since you multiply it by 2 to get 50, you make it the square root of 2. So 5 times √2 is also equal to √50.

Example 2A:Q: Which of these answers is the correct answer for x and y with the information given? Show your work and explain your answer.A. x = 24√3 & y = 24B. x =12√3 & y = 12C. x = 24 & y = 24√3D. x = 12 & y = 12√3A: D. x = 12 & y = 12√3

← 24

x

y

Work: 24 / 2 = 12 x = 12 (12)√3

Explanation: The side you must figure out first is x ,so you must use the formula above. Now you know that 2a (in this case x) = 24 which equals 12 (first part of answer.) You could go on and solve the rest but there is only one answer in which x = 12; the answer is D. To check if its right, plug in 12 into a√3 and get 12√3. 

Example 2B:Q: Using the triangle from Example 2A, figure out which lines are the hypotenuse and the leg. Explain your answer. A: Leg = Line BA     Leg = Line AC     Hypotenuse = Line BC

Explanation:To find the hypotenuse, you must find the side that is diagonal (or opens out to) the right triangle. So, line BC is the hypotenuse. The remaining sides are both the legs.

1st Example

2nd Example