Answer:
- 60°
- 20
- 20/√3
- 20/√3
- 40/√3
Step-by-step explanation:
Given right triangle ABC with A=60°, C=90°, and altitude CD=10, you want the measure of angle BCD, the length of BC, the length of AC from BC, the length of AC from CD, and the length of AB.
1. Angle BCD
All of the triangles in the figure are similar. Any acute angle in a right triangle that is not 30° will be 60°, and vice versa. In ∆BCD, ∠B=30°, so ∠BCD=60°.
2. Side BC
The ratio of sides in a 30-60-90 triangle is 1 : √3 : 2. Given side CD is the short side in ∆BCD, and BC is the long side. It will be double the length of CD.
BC = 2·10 = 20
3. AC from BC
In ∆ABC, AC is the short side, and BC is the middle-length side. This tells you ...
AC = BC/√3
AC = 20/√3
4. AC from CD
In ∆ACD, AC is the hypotenuse, and CD is the middle length side. This tells you ...
AC = CD(2/√3) = 10(2/√3)
AC = 20/√3
5. Side AB
In ∆ABC, AB is the hypotenuse and AC is the short side. AB will be twice the length of AC:
AB = 2·AC = 2(20/√3)
AB = 40/√3
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Additional comment
If you want these numbers with a rational denominator, you can use ...
1/√3 = (√3)/3
