A circle with center O has two parallel chords, AB and CD. The chords form an isosceles trapezoid ABDC where...

1. TRANSLATE the problem information

• Given information:
  • Circle with center O has two parallel chords AB and CD
  • \(\mathrm{AB = 150\text{ cm}, CD = 234\text{ cm}}\)
  • Distance between chords = 144 cm
  • Center O lies between the chords
  • Need to find: radius r

2. INFER the geometric relationships

• Key insight: The perpendicular from the center of a circle to any chord bisects that chord
• This means we can work with half-chord lengths:
  • Half of CD = \(\mathrm{234/2 = 117\text{ cm}}\)
  • Half of AB = \(\mathrm{150/2 = 75\text{ cm}}\)
• The perpendiculars from O to each chord create right triangles

3. VISUALIZE the setup with distances

• Let \(\mathrm{h_1}\) = distance from center O to chord CD
• Let \(\mathrm{h_2}\) = distance from center O to chord AB
• Since O lies between the chords: \(\mathrm{h_1 + h_2 = 144}\)

4. INFER the Pythagorean relationships

• For each chord, we have a right triangle where:
  • Hypotenuse = radius r
  • One leg = distance from center to chord
  • Other leg = half the chord length
• This gives us:
  • For chord CD: \(\mathrm{r^2 = h_1^2 + 117^2}\)
  • For chord AB: \(\mathrm{r^2 = h_2^2 + 75^2}\)

5. SIMPLIFY by setting up equations

• Since both expressions equal \(\mathrm{r^2}\):
  \(\mathrm{h_1^2 + 117^2 = h_2^2 + 75^2}\)
• Rearranging: \(\mathrm{h_2^2 - h_1^2 = 117^2 - 75^2}\)
• Calculate: \(\mathrm{117^2 - 75^2 = 13,689 - 5,625 = 8,064}\)

6. SIMPLIFY using difference of squares

• Factor: \(\mathrm{(h_2 - h_1)(h_2 + h_1) = 8,064}\)
• Substitute \(\mathrm{h_2 + h_1 = 144}\): \(\mathrm{(h_2 - h_1)(144) = 8,064}\)
• Solve: \(\mathrm{h_2 - h_1 = 8,064 \div 144 = 56}\)

7. SIMPLIFY the linear system

• Now we have:
  • \(\mathrm{h_2 + h_1 = 144}\)
  • \(\mathrm{h_2 - h_1 = 56}\)
• Adding these equations: \(\mathrm{2h_2 = 200}\), so \(\mathrm{h_2 = 100}\)
• Substituting back: \(\mathrm{h_1 = 144 - 100 = 44}\)

8. SIMPLIFY to find the radius

• Using \(\mathrm{r^2 = h_2^2 + 75^2}\):
  \(\mathrm{r^2 = 100^2 + 75^2}\)
  \(\mathrm{r^2 = 10,000 + 5,625}\)
  \(\mathrm{r^2 = 15,625}\)
• Taking the square root: \(\mathrm{r = 125}\)

Answer: C. 125

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the perpendicular from center bisects the chord, so they use the full chord lengths (150 and 234) instead of half-lengths (75 and 117) in their Pythagorean theorem setup.

This leads to incorrect equations like \(\mathrm{r^2 = h_1^2 + 234^2}\) and \(\mathrm{r^2 = h_2^2 + 150^2}\), which produces a completely different system and wrong answer. This may lead them to select an answer choice that doesn't match any of the given options, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the initial Pythagorean relationships but make arithmetic errors when calculating \(\mathrm{117^2 - 75^2 = 8,064}\), often getting \(\mathrm{117^2 = 13,689}\) wrong or making errors in the division \(\mathrm{8,064 \div 144 = 56}\).

These calculation errors propagate through the linear system solution, leading to incorrect values for \(\mathrm{h_1}\) and \(\mathrm{h_2}\), and ultimately an incorrect radius. This may lead them to select Choice B (117) if they confuse the half-chord length with the final answer.

The Bottom Line:

This problem requires both strong geometric insight (recognizing the chord bisection property) and careful algebraic manipulation. Students who rush through either the setup or the calculations will likely arrive at incorrect answers.