Problem
M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 6 / 2 = 3 cm.
Diagonals of a parallelogram split each other evenly, so half of the 6 cm diagonal is 3 cm.
4.G.A.2Identify SubproblemsThe crossing point M splits diagonal BD into two equal halves, so MB is exactly half of the full 6 cm diagonal.
BD is made of the two pieces MB and MD, so once those two pieces are known to be equal, each one has to be half of BD.
The piece MB followed by the piece MD makes up the whole diagonal BD, with nothing left out and nothing counted twice.
The two pieces MB and MD are the same length, because M is the point where a parallelogram's diagonals cross and that point sits at the exact middle of each diagonal.
Turn the whole parallelogram a half turn around M: it lands back exactly on itself and corner B lands right where corner D was, so the step from M to B matches the step from M to D.
A half turn about M lays one half of the figure exactly onto the other half, and sliding a shape onto itself this way leaves every length unchanged, so MB comes out equal to MD.
M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 5.4 cm.
Since M halves AC, the piece from M to C is the same 5.4 cm as the piece from A to M.
4.G.A.2Identify SubproblemsIn a parallelogram opposite sides are equal, and BC is opposite AD (not AB), so BC = AD = 7 cm.
Opposite sides of a parallelogram match, so BC copies AD's 7 cm.
4.G.A.2Draw A DiagramPerimeter of triangle BCM = MB + CM + BC = 3 + 5.4 + 7 = 15.4 cm.
Adding the three side lengths gives the distance all the way around the triangle.
4.MD.A.3Identify SubproblemsA parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!