find the length of segment a.
$b = 21\text{ m}$
$c = 7\text{ m}$
$d = 18\text{ m}$
$a = ?\text{ m}$

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Intersecting Secants Theorem", "Circle Geometry" ], "new_concepts": [], "current_concepts": [ "Intersecting Secants Theorem", "Circle Geometry" ] } ## State the Intersecting Secants Theorem Using the *Intersecting Secants Theorem* and *Circle Geometry* knowledge points $$ a \cdot (a + b) = c \cdot (c + d) $$ ## Substitute the given values Using the *Circle Geometry* knowledge point $$ \begin{aligned} &b = 21\ &c = 7\ &d = 18\ &a \cdot (a + 21) = 7 \cdot (7 + 18) \end{aligned} $$ ## Simplify the equation Using the *Circle Geometry* knowledge point $$ \begin{aligned} &a^2 + 21a = 7 \cdot 25\ &a^2 + 21a = 175\ &a^2 + 21a - 175 = 0 \end{aligned} $$ ## Solve the quadratic equation Using the *Circle Geometry* knowledge point $$ \begin{aligned} &a = \frac{-21 \pm \sqrt{21^2 - 4(1)(-175)}}{2(1)}\ &a = \frac{-21 \pm \sqrt{441 + 700}}{2}\ &a = \frac{-21 \pm \sqrt{1141}}{2} \end{aligned} $$ ## Determine the physical length Using the *Circle Geometry* knowledge point $$ \begin{aligned} &a > 0\ &a = \frac{-21 + \sqrt{1141}}{2} \approx 6.39 \end{aligned} $$ Find the length of segment a. $a =$ $\frac{-21 + \sqrt{1141}}{2}$ m (or approximately 6.39 m) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Intersecting Secants Theorem" ] }

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Question

Explanation:

State the Intersecting Secants Theorem

Substitute the given values

Simplify the equation

Solve the quadratic equation

Determine the physical length

Answer: