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();
• `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`