c What is true about triangles where all three sides are congruent?
D
d What type of triangle has a hypotenuse with a length of 5 units and a leg with a length of 3 units?
A
a No, see solution.
B
bSimilar? Yes.
Transformation: Translation, rotation, dilation.
C
cSimilar? Yes.
Transformation: Translation, rotation, dilation.
D
dSimilar? Yes.
Transformation: Translation, rotation, dilation.
Practice makes perfect
a If two shapes are similar, the ratio of corresponding side lengths is the same regardless of which corresponding sides you choose. Also, since relative side lengths are preserved in similar shapes, the shortest sides are corresponding, the longest are corresponding, and so on. We can identify which sides should be corresponding if the triangles are similar.
Again, if these figures are similar the ratio between corresponding sides is the same. Let's investigate this proposition by writing an equation.
10/15? =12/18? =11/17
If the three ratios give the same result, the shapes are similar.
Since the ratio of what would be corresponding sides is not the same, these cannot be similar triangles.
b Both of these triangles are isosceles, which means their legs are corresponding.
Additionally, we see that the angle between the corresponding sides are congruent. This means we can claim similarity by the SAS (Side-Angle-Side Similarity) Condition.
Sequence of Transformations
The triangles have different positions, different orientations, and different sizes. Therefore, to map one onto the other we have to perform a translation, a rotation, and a dilation with a scale factor of 58 if we decide to transform the larger triangle.
c In both triangles the three sides are congruent, which means both of these triangles are equilateral triangles. Equilateral triangles have three congruent angles, each of them 60^(∘). Therefore, these triangles are similar.
Sequence of Transformations
The triangles have different positions, different orientations, and different sizes. Therefore, to map one onto the other we have to perform a translation, a rotation, and a dilation by a scale factor less than 1 if we decide to transform the larger triangle.
d We have two right triangles. In △ TUV the hypotenuse is 5 units and a leg is 3 units, which means this is a 3-4-5 triangle which is a Pythagorean Triple. If we can prove that the legs of △ YXW are multiples of 3 and 4, we know this is a scaled up version of a 3-4-5 triangle. Since 8 and 6 are both multiples of 2, we can scale down the triangle by this factor.
As we can see, the larger triangle is in fact a dilated 3-4-5 triangle, which means they are similar triangles.
Sequence of Transformations
The triangles have different positions, different orientations, and different sizes. Therefore, to map one onto the other, we have to perform a translation, a rotation, and a dilation by a scale factor of 12 if we transform the larger triangle.
