randomSides() triangleAngles( ABC_SIDES ) [ "DEF", "GHI", "Both", "None" ] { "DEF": "<code>DEF</code>", "GHI": "<code>GHI</code>", "Both": "Both", "None": "None" } ANSWERS[randRange(0, 3)] randRange(1, 3)/2 randRange(1, 3)/2 ANSWER === "DEF" || ANSWER === "Both" ? scaleSides(ABC_SIDES, SCALE_DEF) : randomSides(SCALE_DEF, ABC_SIDES) ANSWER === "DEF" || ANSWER === "Both" ? ABC_ANGLES : triangleAngles(DEF_SIDES) ANSWER === "GHI" || ANSWER === "Both" ? scaleSides(ABC_SIDES, SCALE_GHI) : randomSides(SCALE_DEF, ABC_SIDES) ANSWER === "GHI" || ANSWER === "Both" ? ABC_ANGLES : triangleAngles(GHI_SIDES) "\\neq" ABC_SIDES[2] / DEF_SIDES[2] === ABC_SIDES[0] / DEF_SIDES[0] ? "=" : "\\neq" ABC_SIDES[0] / DEF_SIDES[0] === ABC_SIDES[1] / DEF_SIDES[1] ? "=" : "\\neq" ABC_SIDES[2] / GHI_SIDES[2] === ABC_SIDES[0] / GHI_SIDES[0] ? "=" : "\\neq" ABC_SIDES[0] / GHI_SIDES[0] === ABC_SIDES[1] / GHI_SIDES[1] ? "=" : "\\neq" function(){ var tr = new Triangle( [ 2, -1 ], ABC_ANGLES, 5, {} ); tr.labels = {"sides": [ABC_SIDES[2], ABC_SIDES[0], ABC_SIDES[1]], "points" : ["A", "B", "C"] }; tr.rotate( randRange( 0, 360 ) ); tr.boxOut( [ [ [ -4, 1.5 ], [ 10, 1.5 ] ] ], [ 0, -0.5 ] ); return tr; }() function(){ var trA = new Triangle( [ 1, -8 ], DEF_ANGLES, 5*SCALE_DEF, {} ); trA.labels = {"sides": [DEF_SIDES[2], DEF_SIDES[0], DEF_SIDES[1]], "points" : ["D", "E", "F"] }; trA.rotate( randRange( 0, 360 ) ); trA.color = "blue"; trA.boxOut( [ [ [ -1, -10 ], [ -1, 20 ] ] ], [ 0.5, 0 ] ); trA.boxOut( TR.sides, [ 0, -1 ] ); return trA; }() function(){ var trB = new Triangle( [ 8, -6.5 ], GHI_ANGLES, 5*SCALE_GHI, {} ); trB.labels = {"sides": [GHI_SIDES[2], GHI_SIDES[0], GHI_SIDES[1]], "points" : ["G", "H", "I"] }; trB.rotate( randRange( 0, 360 ) ); trB.color = "red"; trB.boxOut( [ [ [ 13, -10 ], [ 13, 20 ] ] ], [ -0.5, 0 ] ); trB.boxOut( TR.sides, [ 0, -1 ] ); trB.boxOut( TR_A.sides, [ 0, -1 ] ); return trB; }()
Which triangles are similar to triangle ABC?
init({ range: [ [-1, 13 ], [ -14, 2.5 ] ], scale: 35 }) TR.draw(); TR.drawLabels(); style({ stroke: "blue", }); TR_A.draw(); TR_A.drawLabels(); style({ stroke: "red", }); TR_B.draw(); TR_B.drawLabels();
ANSWER_DISPLAY[ ANSWER ]
  • DEF
  • GHI
  • Both
  • None

The sides of similar triangles are always proportional. This is known as

\color{orange}{Side-Side-Side (SSS) Similarity}.

First, let's determine whether ABC and DEF are similar.

In triangle DEF, DE = DEF_SIDES[2], EF = DEF_SIDES[0], and FD = DEF_SIDES[1].

In triangle ABC, AB = ABC_SIDES[2], BC = ABC_SIDES[0], and CA = ABC_SIDES[1].

In order for ABC and DEF to be similar:

\dfrac{AB}{\color{blue}{DE}} = \dfrac{BC}{\color{blue}{EF}} = \dfrac{CA}{\color{blue}{FD}}

Substitute in the proper values for each side.

\dfrac{ABC_SIDES[2]}{\color{blue}{DEF_SIDES[2]}} DEF_COMP_1 \dfrac{ABC_SIDES[0]}{\color{blue}{DEF_SIDES[0]}} DEF_COMP_2 \dfrac{ABC_SIDES[1]}{\color{blue}{DEF_SIDES[1]}}

Since not all the proportions are equal, ABC is not similar to DEF.

Next, let's determine whether ABC and GHI are similar.

In triangle GHI, DE = GHI_SIDES[2], EF = GHI_SIDES[0], and FD = GHI_SIDES[1].

In triangle ABC, AB = ABC_SIDES[2], BC = ABC_SIDES[0], and CA = ABC_SIDES[1].

For triangles ABC and GHI to be similar:

\dfrac{AB}{\color{red}{GH}} = \dfrac{BC}{\color{red}{EF}} = \dfrac{CA}{\color{red}{FD}}

Substitute in the proper values for each side.

\dfrac{ABC_SIDES[2]}{\color{red}{GHI_SIDES[2]}} GHI_COMP_1 \dfrac{ABC_SIDES[0]}{\color{red}{GHI_SIDES[0]}} GHI_COMP_2 \dfrac{ABC_SIDES[1]}{\color{red}{GHI_SIDES[1]}}

Since not all the proportions are equal, ABC is not similar to GHI.

DEF is similar to ABC

GHI is similar to ABC

DEF and GHI are similar to ABC

Neither DEF nor GHI are similar to ABC