In the given figure, the diameter of the circle is 3 cm. AB and MN are two diameters such that MN is perpendicular to AB. In addition, CG is perpendicular to AB such that \( AE : EB = 1 : 2 \), and DF is perpendicular to MN such that \( NL : LM = 1 : 2 \). The length of DH in cm is:
Show Hint
In geometry problems involving ratios and right-angle triangles, always apply the Pythagorean theorem and the properties of similar triangles to break down the problem into smaller, solvable parts.
Step 1: Understand the geometric setup The figure is a combination of geometric elements including a circle with two perpendicular diameters, AB and MN, intersecting at the center. Points \( CG \) and \( DF \) are drawn perpendicular to the diameters.
Step 2: Use given ratio information We are given that the ratio \( AE : EB = 1 : 2 \), which implies that the line segment \( AB \) is divided in a ratio of 1:2 at point \( E \). Similarly, the ratio \( NL : LM = 1 : 2 \) divides the segment \( MN \).
Step 3: Apply geometric principles First, note that since \( AE : EB = 1 : 2 \), we can calculate that:
\[
AE + EB = AB = 3 \quad \text{(as \( AB = 3 \))}
\]
\[
AE = 1 \quad \text{and} \quad EB = 2
\]
Thus, \( AB = 3 \) and \( AE = 1 \).
Next, we apply this ratio in the context of triangle \( OLD \), where:
\[
% Option
(OD)^2 = (OL)^2 + (DL)^2
\]
We use this formula to calculate \( DL \).
\[
DL = \sqrt{2} \quad \text{from the relation for triangle \( OLD \)}
\]
Step 4: Solve for DH using the Pythagorean theorem Now we can solve for the length of \( DH \) based on the geometric properties of the figure:
\[
DH = DL - HL = \sqrt{2} - 1
\]
This simplifies to the required result:
\[
DH = \frac{2\sqrt{2} - 1}{2}
\]
Final Answer: The correct answer is (b) \( \frac{2\sqrt{2}-1}{2} \).
