Question

In the diagram given below, \( \angle ABD = \angle CDB = \angle PQD = 90^{\circ} \). If AB:CD = 3:1, the ratio of CD: PQ is

• A. 1 : 0.69
• B. 1 : 0.75
• C. 1 : 0.72
• D. None of the above.

Hint:

AB, PQ and CD are all parallel, so look for two pairs of similar triangles that share a vertex at B and at D. Write PQ in terms of the fraction BQ/BD in both.

Answer 1

The Correct Option is B

Step 1: Understand the figure.
AB, PQ and CD are all perpendicular to the line BD, so they are parallel to each other. The lines AD and BC cross at P.
Let \( CD = 1 \) unit. Then \( AB = 3 \) units, because \( AB:CD = 3:1 \). Let \( PQ = h \) and \( BD = d \), and let \( BQ = x \), so \( QD = d - x \).

Step 2: First pair of similar triangles.
In triangles BPQ and BCD, the angle at B is common and both have a right angle (at Q and at D). So they are similar.
\[ \frac{PQ}{CD} = \frac{BQ}{BD} \quad \Rightarrow \quad \frac{h}{1} = \frac{x}{d} \]

Step 3: Second pair of similar triangles.
In triangles DPQ and DAB, the angle at D is common and both have a right angle (at Q and at B). So they are similar.
\[ \frac{PQ}{AB} = \frac{QD}{BD} \quad \Rightarrow \quad \frac{h}{3} = \frac{d - x}{d} = 1 - \frac{x}{d} \]

Step 4: Combine the two results.
From step 2, \( \frac{x}{d} = h \). Put this in step 3:
\[ \frac{h}{3} = 1 - h \quad \Rightarrow \quad h + \frac{h}{3} = 1 \quad \Rightarrow \quad \frac{4h}{3} = 1 \quad \Rightarrow \quad h = \frac{3}{4} = 0.75 \]
Notice that d has cancelled out, so the answer does not depend on the length BD.

Step 5: Write the ratio.
\( CD : PQ = 1 : 0.75 \), which is option 2.

Step 6: Why the other options fail.
Options 1 and 3 give PQ as 0.69 and 0.72 when CD is 1, but the working gives exactly 0.75, so both are wrong.
Option 4 (None of the above) is wrong because 1 : 0.75 is already listed as option 2.

Final Answer:
PQ is three fourths of CD, so \( CD : PQ = 1 : 0.75 \), which is option 2. \[ \boxed{2} \]