Solve: please include statement and reason
Given: ΔАВС, m∠ACB = 90°
CD ⊥ AB, m∠ACD = 30°
AD = 6 cm.
Find: BD
m∠ACD = 30°
sin 30 = 1/2 = AD / AC = 6 /AC
so AC = 12
tan 60 = CB/12
so CB = 12 tan 60
sin 60 = BD/CB = BD/ (12 tan 60)
BD = 12 sin 60 tan 60 = 12 (.5 sqrt 3)(sqrt 3) = 6*3 = 18
You have a 30-60-90° triangle
You should know the ratio of sides are : 1:√3:2
so CD/6 = √3:1
CD = 6√3
To solve for BD, we can use the properties of right triangles and trigonometric ratios.
Given that ΔABC is a right triangle with ∠ACB = 90°, we can use the properties of right triangles to find the length of CD.
Since CD is perpendicular to AB at point D, we have two right triangles within ΔABC: ΔACD and ΔBCD.
From ΔACD, we know that ∠ACD = 30° and AD = 6 cm.
To find CD, we can use the trigonometric ratio for the sine function:
sin(30°) = Opposite / Hypotenuse
sin(30°) = CD / AD
sin(30°) = CD / 6
CD = 6 * sin(30°)
CD = 3 cm
Now, to find BD, we need to consider the right triangle ΔBCD.
Since ∠ACB = 90°, ∠BCD = 180° - ∠ACB = 180° - 90° = 90°.
In ΔBCD, we can use the Pythagorean Theorem:
BC² = BD² + CD²
BD² = BC² - CD²
From ΔACD, we know that AD = 6 cm and CD = 3 cm. Therefore, BC can be found using the Pythagorean Theorem:
BC² = AD² + CD²
BC² = 6² + 3²
BC² = 36 + 9
BC² = 45
Substituting this value into the equation for BD²:
BD² = 45 - 3²
BD² = 45 - 9
BD² = 36
Taking the square root of both sides:
BD = √36
BD = 6 cm
Therefore, BD = 6 cm.