5.1.2 Medians, Altitudes, and Midsegments in Triangles (continued)
In this example, it is given that XY is a midsegment of triangle ABC . Use the definition of midsegment and the Triangle Midsegment Theorem to find the missing lengths. By the definition of midsegment, X is the midpoint of AB . It follows that XA = XB , by the definition of midpoint. In the figure, it is given that XB = 8. So, XA = 8 as well.
XY is a midsegment. So, by the Triangle Midsegment Theorem, XY is half the length of AC . In the figure, it is given that AC = 20. It follows that XY = (1/2)(20) = 10.
By the definition of midsegment, X is the midpoint of BC . It follows that BY = YC , by the definition of midpoint. In the figure, it is given that YC = 6. So, BY = 6 as well. Here, the measure of an angle is to be found. By the Triangle Midsegment Theorem, midsegment XY is parallel to AC . Notice that PY is a transversal that passes through parallel lines XY and AC . Therefore, the theorems and postulates regarding corresponding, alternate interior, and alternate exterior angles can be applied. Consider angles XYP and YPC . These are alternate interior angles. Therefore, by the Alternate Interior Angles Theorem, m ∠ XYP = m ∠ YPC . It is given that m ∠ XYP = 25 ° . So, m ∠ YPC = 25 ° as well.
266
Made with FlippingBook - Online magazine maker