Question

Josephine is constructing a circle circumscribed about triangle XYZ. She uses the following sequence of steps to construct the circle.

1. She constructs the perpendicular bisectors of two sides.
2. She finds their point of intersection, P.
3. She uses a compass, with the fixed end on X, to draw a circle passing through point P.
Did Josephine make any errors while constructing the circle?
A. Yes, she should have placed the compass with the fixed end on P and drawn a circle through point x.
B. No, she did not make any errors.
C. Yes, she should have constructed the angle bisectors.
D. Yes, she should have found the perpendicular bisectors of all 3 sides.

Answer 1

Josephine made an error in step 3; she should have placed the compass on point P, the circumcenter, to draw the circle.

Step-by-step explanation:

The student is asking about the correct steps for constructing a circle circumscribed about a triangle. Josephine's method has an error in step 3. The correct method is:

1. Construct perpendicular bisectors for each side of the triangle.
2. Find the point of intersection, P, of the perpendicular bisectors. This point P is the circumcenter of the triangle.
3. Place the fixed end of the compass on point P and adjust the width to reach any vertex of the triangle (e.g., point X), and draw the circle. This ensures that the circle will pass through all three vertices of the triangle.

Therefore, the error Josephine made was in step 3 of her construction process. The correct answer is:

A. Yes, she should have placed the compass with the fixed end on P and drawn a circle through point X.

Answer 2

Josephine made a mistake in her construction method for a circumscribed circle. She should have drawn the circle with the fixed end of the compass at point P, intersecting the perpendicular bisectors, not at vertex X.

Step-by-step explanation:

The student is attempting to construct a circle circumscribed about a triangle. Josephine followed the correct initial step of constructing the perpendicular bisectors of two sides of the triangle and finding their point of intersection, P. However, she made an error in the last step.

In step 3, she should have placed the fixed end of the compass on point P, where the perpendicular bisectors intersect, and then drawn a circle passing through any of the triangle's vertices, such as X.

This would ensure that the circle is circumscribed about the triangle, meaning all vertices of the triangle lie on the circle's circumference. Therefore, the correct answer is A: Yes, she should have placed the compass with the fixed end on P and drawn a circle through point X.