Depth · one concept from first principles · step 2 of 4
In the figure, line is used as the axis of symmetry to draw a line-symmetric figure. If triangle ABC is an isosceles triangle, find the area of triangle ABC in .
Reflect B across the axis to B'; symmetry keeps AB' equal to AB and opens the apex to 60 degrees.
A 4th grader knows a line of symmetry makes a mirror copy, so the matching leg and matching angle are equal.
4.G.A.3Draw A DiagramTwo equal legs with a 60 degree apex force both base angles to 60, so BB' is also 12 cm.
Equal legs force equal base angles; with a 60 degree apex all angles turn out to be 60 degrees, the hallmark of an equilateral triangle.
4.MD.C.5Identify SubproblemsThe axis bisects BB', so B stands 12 divided by 2 = 6 cm from AC, which is the height.
The axis is the fold line, so each point sits the same distance from it; half of BB' is exactly how far B is from the base AC.
4.G.A.3Draw A DiagramThe axis cuts the segment joining B to its mirror image exactly in half, so B stands 6 cm from the axis.
The mirror image is where B lands when the figure is folded along the axis, and folding carries B onto it without changing any distance.
That fold swaps B and its image, so the crease has to lie the same distance from each of them, which is the middle of the segment joining them.
The two halves of that segment are matched across the crease, so neither half can be longer than the other.
With base AC = 12 cm and height 6 cm, the area is 12 x 6 divided by 2 = 36 square cm.
Area of any triangle is half the base times the height, the same half-of-a-rectangle idea kids meet in 3rd grade.
3.MD.C.7Identify SubproblemsA line of symmetry is a fold line: the matching point sits the same distance on the other side, so half of that mirror gap is your triangle's height.
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The same shape shows up later on the AMC, as Paper Fold-and-Cut Symmetry. Why the shapes repeat →