In △ABC, m∠ACB=90°, m∠ACD=60°,
CD
is the altitude to
AB
, and BD = 5 cm. Find AD.
(A) 6cm (B) 15cm (C) 20cm (D) 25cm
To find AD, we can use the trigonometric relationship in a right triangle.
In triangle ABC, since angle ACB is a right angle (90°), we can use trigonometric ratios to find the length of AD.
Since angle ACD is 60°, we can find the length of AD using the Sin function.
sin(60°) = opposite/hypotenuse
In this case, the opposite side is AD, and the hypotenuse is CD.
sin(60°) = AD/CD
sin(60°) = AD/5
Using the trigonometric function table, we can find that sin(60°) = √3/2.
√3/2 = AD/5
AD = (5 * √3) / 2
AD = (5√3) / 2
AD ≈ 8.66 cm
Therefore, the approximate length of AD is 8.66 cm.
To find AD, we can use the trigonometric relationship in a right triangle.
Step 1: Identify the right triangle
In triangle ABC, angle ACB is 90 degrees, which means triangle ABC is a right triangle.
Step 2: Determine the relationship between the given angles
In triangle ACD, angle ACD is 60 degrees. Since angle ABC is 90 degrees, angle DCA is 90 - 60 = 30 degrees.
Step 3: Apply trigonometry to find AD
In triangle ACD, we can use the tangent function to find AD. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
In this case, we have:
tan(DCA) = opposite side / adjacent side
tan(30 degrees) = CD / AD
We know that CD is the altitude to AB, meaning it is perpendicular to AB. Therefore, CD and AB are perpendicular, forming a 90-degree angle. This makes CD the opposite side and AD the adjacent side in relation to angle DCA.
Now, we can substitute the values into the equation and solve for AD:
tan(30 degrees) = CD / AD
tan(30 degrees) = 5cm / AD
To find AD, we can rearrange the equation:
AD = 5cm / tan(30 degrees)
Using a calculator, we can find the value of tan(30 degrees) ≈ 0.577.
AD = 5cm / 0.577
AD ≈ 8.66 cm
Therefore, the length of AD is approximately 8.66 cm.
None of the given answer choices match this result. Please check the options or re-evaluate the question.
answer is 15
After you make your sketch, you will realize that your are dealing
with 3 similar triangles with angles 30°, 60°, 90°
with corresponding sides in the ratio of 1 : √3 : 2
So should be easy to see with simple ratios that CD = 5√3 and x = 15
or, ..... you could just use basic trig
In triangle BCD, tan60° = CD/5
√3 = CD/5
CD = 3√5 , as above
Now in triangle ADC, tan60° = x/CD
x = CDtan60°
= (3√5)(√3) = 15 , as above