property: if a tangent and a secant intersect outside a circle, then the length of the tangent squared equals the length of the secant outside the circle times the length of the whole secant.

find the length of each segment. for any answers with a square root, write your answer in the form sqrt(____) with no spaces.

cf =
ce =
cb =

prove:
$ (\quad)^2 = \quad \cdot \quad $
$ (\quad)^2 = \quad \cdot \quad $

simplify both sides of the equation.
$ \quad = \quad $

Question

property: if a tangent and a secant intersect outside a circle, then the length of the tangent squared equals the length of the secant outside the circle times the length of the whole secant.

find the length of each segment. for any answers with a square root, write your answer in the form sqrt(____) with no spaces.

cf =
ce =
cb =

prove:
$ (\quad)^2 = \quad \cdot \quad $
$ (\quad)^2 = \quad \cdot \quad $

simplify both sides of the equation.
$ \quad = \quad $

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Distance Formula in Coordinate Geometry", "Tangent-Secant Theorem" ], "new_concepts": [], "current_concepts": [ "Distance Formula in Coordinate Geometry", "Tangent-Secant Theorem" ] } ## Find coordinates of points from the grid $$ \begin{aligned} &C = (8, 3)\ &F = (3, 6)\ &E = (-2, 9)\ &B = (0, 1) \end{aligned} $$ ## Calculate segment lengths using the distance formula $$ \begin{aligned} &CF = \sqrt{(8-3)^2 + (3-6)^2} = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}\ &CE = \sqrt{(8-(-2))^2 + (3-9)^2} = \sqrt{10^2 + (-6)^2} = \sqrt{100 + 36} = \sqrt{136}\ &CB = \sqrt{(8-0)^2 + (3-1)^2} = \sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68} \end{aligned} $$ ## Formulate and simplify the Tangent-Secant Theorem equation $$ \begin{aligned} &CB^2 = CF \cdot CE\ &(\sqrt{68})^2 = \sqrt{34} \cdot \sqrt{136}\ &68 = \sqrt{4624} = 68 \end{aligned} $$ Property: If a tangent and a secant intersect outside a circle, then the length of the tangent squared equals the length of the secant outside the circle times the length of the whole secant. Find the length of each segment. For any answers with a square root, write your answer in the form sqrt(___) with no spaces. CF = sqrt(34) CE = sqrt(136) CB = sqrt(68) Prove: CB$^2 =$ CF $\cdot$ CE sqrt(68)$^2 =$ sqrt(34) $\cdot$ sqrt(136) Simplify both sides of the equation. 68 = 68 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Tangent-Secant Theorem" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find coordinates of points from the grid

Calculate segment lengths using the distance formula

Formulate and simplify the Tangent-Secant Theorem equation

Answer: