tangent lines

find the perimeter of the polygon if $\angle b \cong \angle d$.

$p = ?\text{ cm}$

Question

tangent lines

find the perimeter of the polygon if $\angle b \cong \angle d$.

$p = ?\text{ cm}$

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": [ "Tangent Segments Theorem", "Circumscribed Polygon" ], "new_concepts": [], "current_concepts": [ "Tangent Segments Theorem", "Circumscribed Polygon" ] } ## Identify tangent segments from vertices Using the *Tangent Segments Theorem* and *Circumscribed Polygon* knowledge points $$ \begin{aligned} &\text{Let the points of tangency on } AB, BC, CD, DA \text{ be } P, Q, R, S \text{ respectively.}\ &AP = AS = 3\text{ cm}\ &BP = BQ = 4\text{ cm}\ &CQ = CR = 5\text{ cm}\ &DR = DS \end{aligned} $$ ## Determine the remaining tangent segment Using the *Tangent Segments Theorem* knowledge point $$ \begin{aligned} &AB = AP + PB = 3 + 4 = 7\text{ cm}\ &BC = BQ + QC = 4 + 5 = 9\text{ cm}\ &\angle B \cong \angle D \implies AB = AD \text{ and } BC = CD \text{ (by symmetry of the circumscribed kite)}\ &AD = AS + SD \implies 7 = 3 + SD \implies SD = 4\text{ cm}\ &DR = DS = 4\text{ cm} \end{aligned} $$ ## Calculate the perimeter of the polygon Using the *Circumscribed Polygon* knowledge point $$ \begin{aligned} &P = AB + BC + CD + DA\ &P = 7 + 9 + (CR + RD) + (AS + SD)\ &P = 7 + 9 + (5 + 4) + (3 + 4)\ &P = 7 + 9 + 9 + 7 = 32\text{ cm} \end{aligned} $$ Find the perimeter of the polygon if $\angle B \cong \angle D$. $P =$ 32 $\text{cm}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Tangent Segments Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify tangent segments from vertices

Determine the remaining tangent segment

Calculate the perimeter of the polygon

Answer: