find the length of each segment.
$x = $
$y = $
$z = $

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem" ], "new_concepts": [ "Intersecting Chords Theorem", "Perpendicular Chord Bisector Theorem" ], "current_concepts": [ "Pythagorean Theorem", "Intersecting Chords Theorem", "Perpendicular Chord Bisector Theorem" ] } ## Find the value of z using the Pythagorean theorem $$ \begin{aligned} z^2 + 5.4^2 &= 10.2^2 \ z^2 + 29.16 &= 104.04 \ z^2 &= 74.88 \ z &\approx 8.65 \end{aligned} $$ ## Find the value of y using the perpendicular bisector property of chords $$ \begin{aligned} y &= 23.2 - z \ y &= 23.2 - 8.65 \ y &= 14.55 \end{aligned} $$ ## Find the value of x using the intersecting chords theorem $$ \begin{aligned} x \cdot 5.4 &= y \cdot z \ 5.4x &= 14.55 \cdot 8.65 \ 5.4x &= 125.8575 \ x &\approx 23.31 \end{aligned} $$ Find the length of each segment. $x =$ 23.31 $y =$ 14.55 $z =$ 8.65 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Intersecting Chords Theorem" ] }

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Explanation:

Find the value of z using the Pythagorean theorem

Find the value of y using the perpendicular bisector property of chords

Find the value of x using the intersecting chords theorem

Answer: