Question

state if the three side lengths form an acute, obtuse, or right triangle.

1. 2, 12, 13

a) acute b) right
c) obtuse

1. 9, 12, 13

a) acute b) right
c) obtuse
find the missing side lengths. leave your answers as radicals in simplest form.
11)
triangle with right angle, 45° angle, leg 7, leg y, hypotenuse x
12)
triangle with right angle, 60° angle, side 4√3, hypotenuse m, leg n
13)
triangle with right angle, 60° angle, leg 2, leg y, hypotenuse x
14)
triangle with right angle, 45° angle, hypotenuse 9√2, leg u, leg v
15)
triangle with right angle, 30° angle, hypotenuse 10, leg m, leg n
extra credit: find the missing side lengths. leave your answers as radicals in simplest form.
16)
triangle with right angle, 30° angle, side 6, leg v, hypotenuse u

Explanation:

Problem 9: Determine the type of triangle with sides 2, 12, 13

Step 1: Identify the longest side

The longest side is 13. Let \( a = 2 \), \( b = 12 \), \( c = 13 \).

Step 2: Apply the Pythagorean inequality

Check \( a^2 + b^2 \) vs \( c^2 \).
\( a^2 + b^2 = 2^2 + 12^2 = 4 + 144 = 148 \)
\( c^2 = 13^2 = 169 \)
Since \( a^2 + b^2 < c^2 \) (\( 148 < 169 \)), the triangle is obtuse.

Step 1: Identify the longest side

The longest side is 13. Let \( a = 9 \), \( b = 12 \), \( c = 13 \).

Step 2: Apply the Pythagorean inequality

Check \( a^2 + b^2 \) vs \( c^2 \).
\( a^2 + b^2 = 9^2 + 12^2 = 81 + 144 = 225 \)
\( c^2 = 13^2 = 169 \)
Since \( a^2 + b^2 > c^2 \) (\( 225 > 169 \)), the triangle is acute.

Step 1: Identify the triangle type

It's a 45 - 45 - 90 triangle, so legs are equal, and hypotenuse \( = \text{leg} \times \sqrt{2} \).

Step 2: Find \( y \)

Since it's a 45 - 45 - 90 triangle, the other leg \( y = 7 \) (legs are equal in a 45 - 45 - 90 triangle).

Step 3: Find \( x \)

Hypotenuse \( x = 7\sqrt{2} \) (hypotenuse formula for 45 - 45 - 90: \( \text{leg} \times \sqrt{2} \)).

Answer:

C) Obtuse

Problem 10: Determine the type of triangle with sides 9, 12, 13