question 7 of 40

given circle $x$ with radius 5 units and chord $ab$ with length 8 units, what is the length of segment $xc$, which bisects the chord?

a. 9
b. 3
c. 6.4
d. 7

Question

question 7 of 40

given circle $x$ with radius 5 units and chord $ab$ with length 8 units, what is the length of segment $xc$, which bisects the chord?

a. 9
b. 3
c. 6.4
d. 7

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Radius of Circle", "Chord Definition", "Perpendicular Bisector of a Chord" ], "new_concepts": [ "Pythagorean Theorem in Circles" ], "current_concepts": [ "Radius of Circle", "Chord Definition", "Perpendicular Bisector of a Chord", "Pythagorean Theorem in Circles" ] } ## Find the length of the bisected chord segment $$ CB = \frac{AB}{2} = \frac{8}{2} = 4 $$ ## Apply the Perpendicular Bisector Theorem $$ XC \perp AB \implies \angle XCB = 90^\circ $$ ## Solve for XC using the Pythagorean theorem $$ \begin{aligned} XC^2 + CB^2 &= XB^2 \ XC^2 + 4^2 &= 5^2 \ XC^2 + 16 &= 25 \ XC^2 &= 9 \ XC &= 3 \end{aligned} $$

A. 9

B. 3

C. 6.4

D. 7

{ "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "X", "color": "#583C87", "withLabel": true } }, { "type": "circle", "params": [[0, 0], 5], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "point", "params": [[-4, 3]], "properties": { "name": "A", "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[4, 3]], "properties": { "name": "B", "color": "#5583F2", "withLabel": true } }, { "type": "segment", "params": [[-4, 3], [4, 3]], "properties": { "strokeColor": "#55DDF2", "strokeWidth": 2 } }, { "type": "point", "params": [[0, 3]], "properties": { "name": "C", "color": "#F2557F", "withLabel": true } }, { "type": "segment", "params": [[0, 0], [0, 3]], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2 } }, { "type": "segment", "params": [[0, 0], [4, 3]], "properties": { "strokeColor": "#B48FF2", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Perpendicular Bisector of a Chord" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the length of the bisected chord segment

Apply the Perpendicular Bisector Theorem

Solve for XC using the Pythagorean theorem

Answer: