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<p>ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is
AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along
ED.</p>
<p>What is the area of the triangle PQR? <mdo:image alt="Cube with triangle inside" src="tri_incube.gif"></mdo:image></p>
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<p>ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2),<br></br>
point Q is 1/3 along GH and point R is 1/3 along ED.</p>
<p class="editorial">To find the area of the triangle PQR, Bithian Heung's solution use Pythagoras' theorem twice and shows that triangle PQR is an equilateral triangle.</p>
<p>$$BQ = \sqrt {BG^2+ GQ^2} = \sqrt {3^2+ 1^2} = \sqrt {10}$$<br></br>
$$PQ = \sqrt {PB^2 + BQ^2} = \sqrt {2^2 + 10} = \sqrt {14}$$</p>
<p>Similarly $QR = RP = \sqrt {14}$ and hence PQR is an equilateral triangle.</p>
<p>It is useful to remember that an equilateral triangle splits into two 30-60-90 degrees triangles. By Pythagoras theorem again the ratio of the sides is 1: $\sqrt 3$ : 2.</p>
<p>Hence the area of the triangle PQR is $\frac {1}{2} (7\sqrt 3)$ square units.</p>
<mdo:image alt="" src="tri_incube.gif"></mdo:image><br></br></mdoxml>
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Trice
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
2D Geometry, Shape and Space
Pythagoras' theorem
2D Geometry, Shape and Space
Mixed triangles
Using, Applying and Reasoning about Mathematics
Visualising