Medians, angle bisectors, and altitudes

{"questions":[{"content":"[[image-1]]<br>[[matcher-1]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/1-01-18.svg"},"matcher-1":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$BM$","$AE$","$BH$","$CH$"]}},"hints":["Find the segment from a vertex to the midpoint of the opposite side. This is a median.","An altitude is perpendicular to the base."]},{"content":"[[image-2]]<br>[[matcher-2]]","widgets":{"image-2":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/2-01-20.svg"},"matcher-2":{"type":"matcher","labels":["Altitude","Angle bisector","Median"],"items":["$ST$","$AD$","$MF$","$TR$"]}},"hints":["Find the segment that forms a right angle with the opposite side or its extension. This is an altitude. In an obtuse triangle, it can lie outside the triangle.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-3]]<br>[[matcher-3]]","widgets":{"image-3":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/3-01-18.svg"},"matcher-3":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$AK$","$KD$","$RF$","$TD$"]}},"hints":["Look for the ray that divides the vertex angle into two equal angles. This is an angle bisector.","A median divides the opposite side into two equal segments."]},{"content":"[[image-4]]<br>[[matcher-4]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/4-01-17.svg"},"matcher-4":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$BK$","$BD$","$BM$","$AB$"]}},"hints":["Notice the isosceles triangle: the segment from the apex to the midpoint of the base is a median, an altitude, and an angle bisector.","An altitude forms an angle of $90^\\circ$ with the base."]},{"content":"[[image-5]]<br>[[matcher-5]]","widgets":{"image-5":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/5-01-12.svg"},"matcher-5":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$KM$","$AK$","$EC$","$CF$"]}},"hints":["A segment joining two points on the sides may be a distractor.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-6]]<br>[[matcher-6]]","widgets":{"image-6":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/1-01-18.svg"},"matcher-6":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$BM$","$AE$","$BH$","$MC$"]}},"hints":["Compare the two segments drawn from the same vertex: one is perpendicular to the opposite side; the other meets its midpoint and is a median.","An altitude forms an angle of $90^\\circ$ with the base."]},{"content":"[[image-7]]<br>[[matcher-7]]","widgets":{"image-7":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/2-01-20.svg"},"matcher-7":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$MF$","$AD$","$ST$","$QT$"]}},"hints":["An altitude can lie outside an obtuse triangle.","A median divides the opposite side into two equal segments."]},{"content":"[[image-8]]<br>[[matcher-8]]","widgets":{"image-8":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/3-01-18.svg"},"matcher-8":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$AK$","$KD$","$RF$","$PQ$"]}},"hints":["Find the right-angle mark in the diagram. The segment from a vertex that meets the opposite side at this right angle is an altitude.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-9]]<br>[[matcher-9]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/4-01-17.svg"},"matcher-9":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$BK$","$BD$","$BM$","$DC$"]}},"hints":["In an isosceles triangle, the segment from the apex to the midpoint of the base is a median, an angle bisector, and an altitude.","An altitude is perpendicular to the base."]},{"content":"[[image-10]]<br>[[matcher-10]]","widgets":{"image-10":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/5-01-12.svg"},"matcher-10":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$KM$","$AK$","$EC$","$AC$"]}},"hints":["Find the segment from a vertex to the midpoint of the opposite side. This is a median.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-11]]<br>[[matcher-11]]","widgets":{"image-11":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/1-01-18.svg"},"matcher-11":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$BM$","$AE$","$BH$","$BC$"]}},"hints":["A segment joining a vertex of a triangle to the midpoint of the opposite side is called a median.","An altitude and the corresponding side of the triangle form an angle of $90^\\circ$."]},{"content":"[[image-12]]<br>[[matcher-12]]","widgets":{"image-12":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/2-01-20.svg"},"matcher-12":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$MF$","$AD$","$ST$","$SR$"]}},"hints":["Notice the right-angle mark: an altitude may meet an extension of a side, so part of the segment can lie outside the triangle.","A median divides the opposite side into two equal segments. Matching tick marks indicate this in the diagram."]},{"content":"[[image-13]]<br>[[matcher-13]]","widgets":{"image-13":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/3-01-18.svg"},"matcher-13":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$AK$","$KD$","$RF$","$RP$"]}},"hints":["Find the angle with two matching arcs. The segment dividing it is an angle bisector.","A median divides the opposite side into two equal segments."]},{"content":"[[image-14]]<br>[[matcher-14]]","widgets":{"image-14":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/4-01-17.svg"},"matcher-14":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$BK$","$BD$","$BM$","$AD$"]}},"hints":["In an isosceles triangle, the segment from the apex to the midpoint of the base is a median, an angle bisector, and an altitude.","An altitude and the base form a right angle."]},{"content":"[[image-15]]<br>[[matcher-15]]","widgets":{"image-15":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/5-01-12.svg"},"matcher-15":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$KM$","$AK$","$EC$","$CF$"]}},"hints":["Identify any segment that joins two points on the sides; it may be a distractor.","An altitude forms an angle of $90^\\circ$ with the opposite side, while a median divides that side into two equal segments."]}]}

{"questions":[{"content":"[[image-26]][[matcher-33]]","widgets":{"image-26":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/6-01-10.svg"},"matcher-33":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$DH$","$MN$","$BF$","$QH$"]}},"hints":["A median divides the opposite side into two equal segments.","An altitude forms an angle of $90^\\circ$ with the corresponding side."]},{"content":"[[image-223]][[matcher-236]]","widgets":{"image-223":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/7-01-7.svg"},"matcher-236":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$FR$","$KN$","$CB$","$BD$"]}},"hints":["Notice that an altitude may meet an extension of a side, placing part of the segment outside the triangle.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-509]][[matcher-530]]","widgets":{"image-509":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/8-01-7.svg"},"matcher-530":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$CK$","$CH$","$AM$","$BC$"]}},"hints":["Equal lengths or angles are usually marked with matching ticks or arcs in a diagram.","Equal angles indicate an angle bisector, a right angle indicates an altitude, and equal segments on a side indicate a median."]},{"content":"[[image-939]][[matcher-968]]","widgets":{"image-939":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/9-01-7.svg"},"matcher-968":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$AK$","$BM$","$AF$","$MC$"]}},"hints":["An altitude is perpendicular to the opposite side or its extension.","A median meets the midpoint of the opposite side."]},{"content":"[[image-1500]][[matcher-1537]]","widgets":{"image-1500":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/10-01-6.svg"},"matcher-1537":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$CM$","$ZW$","$PT$","$XY$"]}},"hints":["Look for the right-angle mark: it identifies an altitude.","Matching tick marks on a side show that the point divides it into equal segments. The segment from the opposite vertex to this point is a median."]},{"content":"[[image-2001]][[matcher-2002]]","widgets":{"image-2001":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/10-01-6.svg"},"matcher-2002":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$CM$","$ZW$","$PT$","$XY$"]}},"hints":["An altitude is always perpendicular to the opposite side or its extension. A small square marks the right angle in the diagram.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-2003]][[matcher-2004]]","widgets":{"image-2003":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/6-01-10.svg"},"matcher-2004":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$DH$","$MN$","$BF$","$QH$"]}},"hints":["A median joins a vertex to the midpoint of the opposite side.","A segment that divides an angle into two equal angles is an angle bisector."]},{"content":"[[image-2005]][[matcher-2006]]","widgets":{"image-2005":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/9-01-7.svg"},"matcher-2006":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$AK$","$BM$","$AF$","$MC$"]}},"hints":["Look for the segment from a vertex to the midpoint of the opposite side. This is a median.","A perpendicular segment from a vertex to the opposite side or its extension is an altitude."]},{"content":"[[image-2007]][[matcher-2008]]","widgets":{"image-2007":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/7-01-7.svg"},"matcher-2008":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$FR$","$KN$","$CB$","$BD$"]}},"hints":["An angle bisector divides an angle into two equal angles, shown by matching arcs at the vertex.","An altitude may meet an extension of a side. Check where the right angle is marked."]},{"content":"[[image-2009]][[matcher-2010]]","widgets":{"image-2009":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/8-01-7.svg"},"matcher-2010":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$CK$","$CH$","$AM$","$BC$"]}},"hints":["A median meets the opposite side at a point that divides it into equal segments.","An altitude is marked by a right angle where it meets the corresponding side or its extension."]},{"content":"[[image-2011]][[matcher-2012]]","widgets":{"image-2011":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/10-01-6.svg"},"matcher-2012":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$CM$","$ZW$","$PT$","$XY$"]}},"hints":["Find the segment with the $90^\\circ$ mark. This is an altitude.","Compare the remaining segments: the one from a vertex to the midpoint of the opposite side is a median."]},{"content":"[[image-2013]][[matcher-2014]]","widgets":{"image-2013":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/6-01-10.svg"},"matcher-2014":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$DH$","$MN$","$BF$","$QH$"]}},"hints":["An altitude is a perpendicular segment from a vertex to the opposite side or its extension.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-2015]][[matcher-2016]]","widgets":{"image-2015":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/7-01-7.svg"},"matcher-2016":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$FR$","$KN$","$CB$","$BD$"]}},"hints":["If a point divides a side into two equal segments, the segment from the opposite vertex to that point is a median.","An angle bisector divides an angle into two equal angles."]},{"content":"[[image-2017]][[matcher-2018]]","widgets":{"image-2017":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/9-01-7.svg"},"matcher-2018":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$AK$","$BM$","$AF$","$MC$"]}},"hints":["An altitude forms a right angle with the corresponding side of the triangle or its extension.","A median starts at a vertex and divides the opposite side into two equal segments."]},{"content":"[[image-2019]][[matcher-2020]]","widgets":{"image-2019":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/02/8-01-7.svg"},"matcher-2020":{"type":"matcher","labels":["Median","Angle bisector","Altitude"],"items":["$CK$","$CH$","$AM$","$BC$"]}},"hints":["Check the marks in the diagram: equal angles indicate an angle bisector, while a right angle indicates an altitude.","A median joins a vertex to the midpoint of the opposite side, where matching tick marks show equal lengths."]}]}