Question

vance wants to construct a circle tangent to all three sides of the acute, scalene triangle lmn using the following steps. 1. he will draw altitudes from vertex l and vertex m, and mark their intersection point as o. 2. he will draw the perpendicular from point o to side mn, and mark the intersection point as p. 3. he will draw the circle centered at point o which will pass through point p. which part of vance’s plan requires revision? a. vance should have found the intersection of two perpendicular bisectors of triangle lmn instead of two altitudes. b. vance should have constructed all three altitudes instead of only constructing two altitudes. c. vance should have found the intersection of two angle bisectors of triangle lmn instead of two altitudes. d. vance should have used the compass to draw a circle through point n instead of point p.

Explanation:

A circle tangent to all three sides of a triangle is the incircle. The center of the incircle (incenter) is the intersection of the triangle's angle bisectors, not altitudes (which intersect at the orthocenter, the center of the circumscribed circle for right triangles, not the incircle). Perpendicular bisectors find the circumcenter (for the circumcircle passing through vertices), so option A is incorrect. Drawing two altitudes is enough to find their intersection, so B is wrong. Using point N would not create a circle tangent to all sides, so D is incorrect.

Answer:

C. Vance should have found the intersection of two angle bisectors of triangle LMN instead of two altitudes.