$$ \therefore \mathrm{m}(\angle \mathrm{DME})=360^{\circ}-\left(90^{\circ}+90\right. $$ TRY 5 by Yourself In the opposite figure : $$ \overline{\mathrm{BC}} $$ is a diameter of the circle M , $$ \overline{\mathrm{MD}} \perp \overline{\mathrm{AB}} $$ where $$ \overline{\mathrm{MD}} \cap \overline{\mathrm{AB}}=\{\mathrm{D}\} $$ , $$ \mathrm{m}(\angle \mathrm{C})=40^{\circ} $$ and $$ \mathrm{AC}=8 \mathrm{~cm} $$. Find: (2) The length of $$ \overline{\mathrm{MD}} $$ $$ 1 \mathrm{~m}(\angle \mathrm{DMB}) $$

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UpStudy AI · Accepted Answer

To solve the problem, we need to find the length of $$ \overline{MD} $$ and the measure of angle $$ \angle DMB $$. Let's break down the information given and solve it step by step.

### Known Conditions:
1. $$ \overline{BC} $$ is a diameter of circle $$ M $$.
2. $$ \overline{MD} \perp \overline{AB} $$ at point $$ D $$.
3. $$ m(\angle C) = 40^\circ $$.
4. $$ AC = 8 \, \text{cm} $$.

### Step 1: Find $$ m(\angle DMB) $$

Since $$ \overline{BC} $$ is a diameter, by the inscribed angle theorem, we know that:
$$
m(\angle BMC) = 90^\circ
$$
This is because any angle inscribed in a semicircle is a right angle.

Now, we can find $$ m(\angle DMB) $$:
$$
m(\angle DMB) = m(\angle BMC) - m(\angle C)
$$
Substituting the known values:
$$
m(\angle DMB) = 90^\circ - 40^\circ = 50^\circ
$$

### Step 2: Find the length of $$ \overline{MD} $$

To find the length of $$ \overline{MD} $$, we can use the right triangle $$ \triangle ACD $$. Since $$ \overline{MD} \perp \overline{AB} $$, $$ D $$ is the foot of the perpendicular from $$ M $$ to $$ AB $$.

Using the sine function in triangle $$ ACD $$:
$$
\sin(m(\angle C)) = \frac{MD}{AC}
$$
Substituting the known values:
$$
\sin(40^\circ) = \frac{MD}{8}
$$
Now, we can solve for $$ MD $$:
$$
MD = 8 \cdot \sin(40^\circ)
$$

Now, let's calculate $$ MD $$ using the sine value for $$ 40^\circ $$.

### Calculation of $$ MD $$

Using the sine value:
$$
\sin(40^\circ) \approx 0.6428
$$
Now substituting this value:
$$
MD = 8 \cdot 0.6428 \approx 5.1424 \, \text{cm}
$$

### Final Results
1. $$ m(\angle DMB) = 50^\circ $$
2. $$ \overline{MD} \approx 5.14 \, \text{cm} $$

Thus, the final answers are:
- $$ m(\angle DMB) = 50^\circ $$
- Length of $$ \overline{MD} \approx 5.14 \, \text{cm} $$
$$ m(\angle DMB) = 50^\circ $$ and $$ \overline{MD} \approx 5.14 \, \text{cm} $$.

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