Respuesta :

aachen

Answer:

18 cm

Step-by-step explanation:

Given: In the right [tex]\Delta \text{ABC}[/tex], CD is the altitude to the hypotenuse AB and [tex]m\angle \text{ABC}=30^{\circ}[/tex], [tex]\text{BD}=54[/tex] cm.

To find: AD

Solution: Consider the figure in the  attached file.

In [tex]\Delta \text{BDC}[/tex]

[tex]\text{cos\:30}^{\circ}=\frac{54}{BC}[/tex]

[tex]\frac{\sqrt{3} }{2}=\frac{54}{BC}[/tex]

[tex]\text{BC}=36\sqrt{3}[/tex]

Now, In [tex]\Delta \text{ACB}[/tex]

[tex]\text{cos\:30}^{\circ}=\frac{BC}{AB}[/tex]

[tex]\frac{\sqrt{3} }{2}=\frac{36\sqrt{3}}{AB}[/tex]

[tex]\text{AB}=36\sqrt{3}\times\frac{2}{\sqrt{3} }[/tex]

[tex]\text{AB}=72[/tex]

Now, [tex]\text{AB}=\text{AD}+\text{BD}[/tex]

[tex]\text{AB}=\text{AD}+54[/tex]

[tex]72=\text{AD}+54[/tex]

[tex]\text{AD}=72-54[/tex]

[tex]\text{AD}=18[/tex] cm

Hence, [tex]\text{AD}=18[/tex] cm.

Ver imagen aachen
MrRoyal

The sides of a right-angled triangle can be calculated using trigonometry ratios.

The length of AD is 18 cm

Given that:

[tex]\angle ABC = 30^o[/tex]

[tex]BD = 54cm[/tex]

See attachment for [tex]\triangle ABC[/tex]

First, we calculate CD using the tangent ratio

[tex]\tan(\theta) = \frac{Opposite}{Adjacent}[/tex]

So, we have:

[tex]\tan(30) = \frac{CD}{54}[/tex]

Make CD the subject

[tex]CD = 54 \times \tan(30)[/tex]

[tex]CD = 54\tan(30)[/tex]

Next, AD will be calculated using tangent ratio

[tex]\tan(\theta) = \frac{Opposite}{Adjacent}[/tex]

So, we have:

[tex]\tan(60) = \frac{CD}{AD}[/tex]

Make AD the subject

[tex]AD = \frac{CD}{\tan(60)}[/tex]

Substitute [tex]CD = 54\tan(30)[/tex]

[tex]AD = \frac{54\tan(30)}{\tan(60)}[/tex]

[tex]AD = 18[/tex]

Hence, side length AD is 18 cm

Read more about right-angled triangles at:

https://brainly.com/question/3770177

Ver imagen MrRoyal