Perpendicular from Center to Chord (Motivate) MCQs Quiz | Class 9
This Class IX Mathematics (Code 041) quiz focuses on Unit IV: Geometry, specifically the topic of Perpendicular from Center to Chord. Test your understanding of how the perpendicular bisects the chord and its converse theorem. Click ‘Submit Quiz’ to check your score and ‘Download Answer PDF’ to save your results for revision.
Understanding Perpendiculars from Center to Chord
In the geometry of circles, the relationship between the center of a circle and its chords is governed by two fundamental theorems. These concepts are crucial for solving problems involving lengths, distances, and radii in Class 9 Mathematics.
1. The Perpendicular Theorem
Statement: The perpendicular drawn from the center of a circle to a chord bisects the chord.
This means if you have a circle with center O and a chord AB, and you draw a line OM perpendicular to AB (where M lies on AB), then M is the midpoint of AB. Consequently, AM equals MB.
2. The Converse Theorem
Statement: The line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
This is the reverse logic. If M is the midpoint of chord AB, then the line segment OM joining the center O to M is perpendicular to the chord AB (angle OMA is 90 degrees).
Mathematical Application (Pythagoras Theorem)
Since the perpendicular creates a right-angled triangle (Triangle OMA), we frequently use the Pythagoras theorem to find missing values:
OA2 = OM2 + AM2
Where:
- OA (r): Radius of the circle (Hypotenuse)
- OM (d): Perpendicular distance from center to chord
- AM (l): Half the length of the chord
Quick Revision Summary
| Component | Description | Formula Relationship |
|---|---|---|
| Perpendicular | Line from center to chord at 90 degrees | Bisects the chord (splits it in half) |
| Bisector Line | Line joining center to midpoint of chord | Is perpendicular to the chord (90 degrees) |
| Radius calculation | Distance from center to endpoint | r = square root of (d2 + l2) |
| Chord calculation | Total length of the segment inside circle | Chord Length = 2 * square root of (r2 – d2) |
Extra Practice Questions
- If the radius is 10 cm and the chord length is 12 cm, find the distance from the center. (Answer: 8 cm)
- A chord is at a distance of 5 cm from the center of a circle of radius 13 cm. Find the length of the chord. (Answer: 24 cm)
- In a circle, the perpendicular bisector of a chord always passes through which point? (Answer: The Center)
- Calculate the length of the longest chord of a circle with radius 7 cm. (Answer: 14 cm)
- If a chord passes through the center, what is its distance from the center? (Answer: 0)
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