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Why is Triangle BDC Equilateral in Circle Construction Problem?
Mathematics
Grade 10 (Junior High School)
Question Content
Select the correct answer: Amy constructed this figure by using a compass to draw circle C and then a straightedge to draw diameter AB through point C. Then, with the compass still set equal to the radius of the circle, Amy drew two arcs centered at point B and labeled the points of intersection D and E. She used the straightedge to draw chords AD, AE, and DE. Amy notes that arcs ADB and AEB are semicircles and that △BDC is equilateral. Because ∠BCD is a central angle, she can say that m$\overset{\frown}{BD}$ = 60°. She uses these facts to determine m$\overset{\frown}{AD}$ = m$\overset{\frown}{DE}$ = m$\overset{\frown}{AE}$ = 120°. Finally, she concludes △ADE is equilateral because congruent arcs are intercepted by congruent chords. Why is △BDC is equilateral?
Correct Answer
D
Detailed Solution Steps
1
Step 1: Recall the properties of the circle and compass construction. The compass was set to the radius of circle C when drawing arcs from point B, so BD = BC (radius of the circle).
2
Step 2: Identify that BC and CD are both radii of circle C, so BC = CD = radius of the circle.
3
Step 3: Combine the equal lengths: BD = BC = CD. A triangle with all three sides equal in length is an equilateral triangle, which means the distance between all three vertices (B, D, C) is equal to the radius of the circle.
Knowledge Points Involved
1
Equilateral Triangle Definition and Properties
An equilateral triangle is a triangle where all three sides are of equal length, and all three internal angles are 60°. This definition is used to confirm a triangle is equilateral when all side lengths are proven equal.
2
Circle Radius Properties
All radii of the same circle are equal in length. This is a fundamental circle property, used here to establish that BC and CD are equal, as they are both radii of circle C.
3
Compass Construction for Equal Lengths
When a compass is set to a fixed length (the circle radius in this case) and used to draw an arc from a point on the circle, the distance from the center point to the intersection point of the arc and circle equals the compass length (circle radius). This gives BD = BC in the problem.
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