4 In the diagram, $$ O A=1 \mathrm{~cm} $$, angle $$ A O B= $$ angle $$ B O C= $$ angle $$ C O D=30^{\circ} $$ and angle $$ O A B= $$ angle $$ O B C= $$ angle $$ O C D=90^{\circ} $$. (i) Find the length of $$ O D $$ giving your answer in the form $$ a \sqrt{3} $$.

Question

UpStudy AI · Accepted Answer

To find the length of $$ O D $$, we can analyze the geometry of the problem based on the given angles and lengths.

1. **Understanding the Geometry**:
   - We have a point $$ O $$ and points $$ A, B, C, D $$ arranged in a specific way.
   - $$ O A = 1 \, \text{cm} $$.
   - The angles $$ A O B $$, $$ B O C $$, and $$ C O D $$ are all $$ 30^\circ $$.
   - The angles $$ O A B $$, $$ O B C $$, and $$ O C D $$ are all $$ 90^\circ $$.

2. **Finding Lengths**:
   - Since $$ O A = 1 \, \text{cm} $$ and $$ \angle O A B = 90^\circ $$, we can find the coordinates of point $$ A $$ as $$ (1, 0) $$.
   - Point $$ B $$ is directly above point $$ A $$ at a right angle, so its coordinates will be $$ (1, 1) $$.
   - The angle $$ A O B = 30^\circ $$ means that the line $$ O B $$ makes a $$ 30^\circ $$ angle with the horizontal line $$ O A $$.

3. **Using Trigonometry**:
   - The length $$ O B $$ can be calculated using the sine and cosine of $$ 30^\circ $$:
     $$
     O B = O A \cdot \frac{1}{\cos(30^\circ)} = 1 \cdot \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \, \text{cm}
     $$
   - The vertical component (height) from $$ O $$ to $$ B $$ is:
     $$
     h = O B \cdot \sin(30^\circ) = \frac{2}{\sqrt{3}} \cdot \frac{1}{2} = \frac{1}{\sqrt{3}} \, \text{cm}
     $$

4. **Finding $$ O C $$**:
   - The angle $$ B O C = 30^\circ $$ means that the length $$ O C $$ can be calculated similarly:
     $$
     O C = O B \cdot \frac{1}{\cos(30^\circ)} = \frac{2}{\sqrt{3}} \cdot \frac{1}{\frac{\sqrt{3}}{2}} = \frac{4}{3} \, \text{cm}
     $$
   - The vertical component from $$ O $$ to $$ C $$ is:
     $$
     h_C = O C \cdot \sin(30^\circ) = \frac{4}{3} \cdot \frac{1}{2} = \frac{2}{3} \, \text{cm}
     $$

5. **Finding $$ O D $$**:
   - The angle $$ C O D = 30^\circ $$ means that the length $$ O D $$ can be calculated as:
     $$
     O D = O C \cdot \frac{1}{\cos(30^\circ)} = \frac{4}{3} \cdot \frac{1}{\frac{\sqrt{3}}{2}} = \frac{8}{3\sqrt{3}} \, \text{cm}
     $$

6. **Final Calculation**:
   - To express $$ O D $$ in the form $$ a \sqrt{3} $$:
     $$
     O D = \frac{8}{3\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{8\sqrt{3}}{9} \, \text{cm}
     $$

Thus, the length of $$ O D $$ is $$ \frac{8\sqrt{3}}{9} \, \text{cm} $$.
The length of $$ O D $$ is $$ \frac{8\sqrt{3}}{9} $$ cm.

4 In the diagram, ,
angle angle angle
and angle angle angle .
(i) Find the length of giving your
answer in the form .

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Geometry Questions

4 In the diagram, , angle angle angle and angle angle angle . (i) Find the length of giving your answer in the form .