question 4 of 25

in the diagram below, $\overline{bc}$ is an altitude of $\delta abd$. to the nearest whole unit, what is the length of $\overline{cd}$?

a. 23
b. 20
c. 18
d. 15

Question

question 4 of 25

in the diagram below, $\overline{bc}$ is an altitude of $\delta abd$. to the nearest whole unit, what is the length of $\overline{cd}$?

a. 23
b. 20
c. 18
d. 15

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Mean Theorems", "Similar Right Triangles" ], "new_concepts": [], "current_concepts": [ "Geometric Mean Theorems", "Similar Right Triangles", "Altitude of a Triangle" ] } ## Identify the given geometric relationships Using the *Similar Right Triangles* knowledge point $$ \begin{aligned} &\angle ABD = 90^\circ\ &\overline{BC} \perp \overline{AD}\ &\triangle BCD \sim \triangle ACB \end{aligned} $$ ## Apply the geometric mean theorem Using the *Geometric Mean Theorems* knowledge point $$ \begin{aligned} &BC^2 = CD \cdot AC\ &32^2 = CD \cdot 44 \end{aligned} $$ ## Solve for the unknown length Using the *Geometric Mean Theorems* knowledge point $$ \begin{aligned} &1024 = CD \cdot 44\ &CD = \frac{1024}{44} \approx 23.27 \end{aligned} $$ ## Round to the nearest whole unit Using the *Geometric Mean Theorems* knowledge point $$ CD \approx 23 $$

(A) 23

(B) 20

(C) 18

(D) 15

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Mean Theorems" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given geometric relationships

Apply the geometric mean theorem

Solve for the unknown length

Round to the nearest whole unit

Answer: