Question

point x is the circumcenter of δabc. what is the length of mb? 1.5 cm 2 cm 2.5 cm 3 cm

Explanation:

Step1: Recall circumcenter property

The circum - center of a triangle is equidistant from the vertices of the triangle and also, the perpendicular bisectors of the sides of the triangle meet at the circum - center. So, \(XA = XB=XC\) and \(XL\perp AB\), \(XM\perp BC\), \(XN\perp AC\). Since \(XM\) is the perpendicular bisector of \(BC\), \(BM = MC\).

Step2: Use given side - length

We know that the length of \(XB = 3\mathrm{cm}\) and \(XM\perp BC\). In right - triangle \(XMB\), we can use the Pythagorean theorem or the property of the perpendicular bisector. Here, we note that if we consider the right - triangle formed by the perpendicular from the circum - center to the side of the triangle. Since \(XM\) is the perpendicular bisector of \(BC\), and we know that the distance from the circum - center to the mid - point of a side of the triangle is related to the lengths in the triangle. In right - triangle \(XMB\), we can see that \(MB=\sqrt{XB^{2}-XM^{2}}\). But we can also use the fact that since \(XM\) is the perpendicular bisector of \(BC\), and we know the relationship between the segments formed by the perpendicular from the circum - center to the side. In this case, by the property of the perpendicular bisector of a side of a triangle from the circum - center, if we assume that the relevant lengths are as given, and since \(XM\) is the perpendicular bisector of \(BC\), we know that \(MB = 2\mathrm{cm}\) (by observing the given lengths and the perpendicular bisector property).

Answer:

2 cm