Question

the figure below shows concentric circles, both centered at o.

• chord xy is tangent to the smaller circle.
• the radius of the larger circle is 15 cm.
• the radius of the smaller circle is 12 cm.

on friday’s unit 5 quiz, 20% of mr. dunton’s honors math 3 students calculated the length of chord xy as 9 cm. what misconception or error in reasoning may have caused this result?

Explanation:

Step1: Recall the property of tangent and chord

Since \( XY \) is tangent to the smaller circle, let the point of tangency be \( T \). Then \( OT \perp XY \) (a radius is perpendicular to the tangent at the point of contact), and \( OT \) is the radius of the smaller circle (\( OT = 12 \, \text{cm} \)), and \( OX \) is the radius of the larger circle (\( OX = 15 \, \text{cm} \)). Also, \( T \) bisects \( XY \) (a perpendicular from the center to a chord bisects the chord), so \( XY = 2XT \).

Step2: Apply the Pythagorean theorem

In right triangle \( OTX \), we can find \( XT \) using the Pythagorean theorem: \( XT=\sqrt{OX^{2}-OT^{2}} \). Substituting \( OX = 15 \) and \( OT = 12 \), we get \( XT=\sqrt{15^{2}-12^{2}}=\sqrt{225 - 144}=\sqrt{81} = 9 \, \text{cm} \). Wait, but the length of \( XY \) should be \( 2\times XT \), so \( XY = 2\times9=18 \, \text{cm} \).

Step3: Identify the error

The students probably forgot to double the length of \( XT \) to get the length of the chord \( XY \). They used \( XT \) (which is 9 cm) as the length of \( XY \) instead of multiplying it by 2.

Answer:

The students likely forgot that the perpendicular from the center to the chord bisects the chord, so they used the length of the segment from the point of tangency to one end of the chord (\( XT = 9 \, \text{cm} \)) as the length of the entire chord \( XY \) instead of doubling it (since \( XY = 2XT \), the correct length should be \( 18 \, \text{cm} \)).