Question

quadrilateral bcde is inscribed in circle a as shown. \\( \overline { b d } \\) divides the quadrilateral into two triangles, \\( \triangle b c d \\) and \\( \triangle b e d \\). which statement is true about the triangles?
a. the angle bisectors and the perpendicular bisectors for both triangles intersect at the same point
b. the angle bisectors of \\( \triangle b c d \\) intersect at the same point as those of \\( \triangle b e d \\).
c. the perpendicular bisectors of \\( \triangle b c d \\) intersect at the same point as those of \\( \triangle b e d \\).
d. the angle bisectors of \\( \triangle b c d \\) intersect at the same point as the perpendicular bisectors of \\( \triangle b e d \\)

Explanation:

Step1: Recall the property of perpendicular bisectors

The perpendicular bisectors of a triangle intersect at the circum - center (the center of the circum - circle). Since quadrilateral \(BCDE\) is inscribed in circle \(A\), points \(B\), \(C\), \(D\), \(E\) lie on the circle. For \(\triangle BCD\) and \(\triangle BED\), the circum - center (the point of intersection of perpendicular bisectors) is the center of the circle in which the triangles are inscribed. Here, the circle is circle \(A\).

Step2: Analyze each option

• Option A:

The angle bisectors and perpendicular bisectors are different. The perpendicular bisectors of a triangle are related to the circum - circle (circum - center), while angle bisectors are related to the in - circle (in - center). So, this option is incorrect.

• Option B:

The angle bisectors of a triangle intersect at the in - center. Since the two triangles \(\triangle BCD\) and \(\triangle BED\) are inscribed in the same circle (circum - circle), the perpendicular bisectors (not angle bisectors) intersect at the same point (the center of the circum - circle, which is point \(A\)). So, this option is incorrect.

• Option C:

The perpendicular bisectors of a triangle intersect at the circum - center. Since \(\triangle BCD\) and \(\triangle BED\) are inscribed in the same circle (circle \(A\)), the perpendicular bisectors of \(\triangle BCD\) and \(\triangle BED\) intersect at the center of the circle (point \(A\)).

• Option D:

The angle bisectors of a triangle intersect at the in - center and the perpendicular bisectors intersect at the circum - center. These are two different points (except in the case of an equilateral triangle, which is not indicated here). So, this option is incorrect.

Answer:

C. The perpendicular bisectors of \(\triangle BCD\) intersect at the same point as those of \(\triangle BED\)