Similar Triangles - Trigonometry
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Which of the following is not a theorem to prove that triangles are similar?
Which of the following is not a theorem to prove that triangles are similar?
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ASA (Angle Side Angle) is a theorem to prove triangle congruency.
In this case, we only need two angles to prove that two triangles are similar, so the last side in ASA is unnecessary for this question.
For this purpose, we use the theorem AA instead.
ASA (Angle Side Angle) is a theorem to prove triangle congruency.
In this case, we only need two angles to prove that two triangles are similar, so the last side in ASA is unnecessary for this question.
For this purpose, we use the theorem AA instead.
What does the scale factor of a dilation need to be to ensure that triangles are not only similar but also congruent?
What does the scale factor of a dilation need to be to ensure that triangles are not only similar but also congruent?
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The scale factor of a dilation tells us what we multiply corresponding sides by to get the new side lengths. In this case, we want these lengths to be the same to get congruent triangles. Thus, we must be looking for the multiplicative identity, which is 1.
The scale factor of a dilation tells us what we multiply corresponding sides by to get the new side lengths. In this case, we want these lengths to be the same to get congruent triangles. Thus, we must be looking for the multiplicative identity, which is 1.
One triangle has side measures 2, 4, and 5. Another has sides 4, 8, and 10. Are these triangles similar?
One triangle has side measures 2, 4, and 5. Another has sides 4, 8, and 10. Are these triangles similar?
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To determine if the triangles are similar, set up a proportion.
2/4 = 4/8 = 5/10
When we do this, we cross multiply to get a true statement.
Or, we can find the scale factor.
Since the scale factor is 2 for all three lengths, it becomes clear that these triangles are similar.
To determine if the triangles are similar, set up a proportion.
2/4 = 4/8 = 5/10
When we do this, we cross multiply to get a true statement.
Or, we can find the scale factor.
Since the scale factor is 2 for all three lengths, it becomes clear that these triangles are similar.
One triangle has side measures 
, 
, and 
. Another has side lengths 
, 
, and 
. Are these triangles similar?
One triangle has side measures
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Two triangles are similar if and only if their side lengths are proportional.
In this case, two of the sides are proportional, leading us to a scale factor of 2.
However, with the last side, 
which is not our side length.
Thus, these pair of sides are not proportional and therefore our triangles cannot be similar.
Two triangles are similar if and only if their side lengths are proportional.
In this case, two of the sides are proportional, leading us to a scale factor of 2.
However, with the last side,
Thus, these pair of sides are not proportional and therefore our triangles cannot be similar.
Which of the following triangles are similar?
Which of the following triangles are similar?
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We must remember that there are three ways to prove triangles are similar.
- At least two angles in one triangle are congruent to angles in another (AA)
- All three pairs of corresponding sides are proportional (SSS)
- Two pairs of corresponding sides are proportional and the angles between those sides are congruent (SAS)
Comparing triangles I and II, we only have one angle and two sides in trinagle II, so attempting to use either AA or SSS for similarity will not work, leaving SAS as the only option. If we compare the two given sides in each triangle, we notice that the ratio of the longer side in triangle I to the longer side in triangle II is
The ratio of the shorter sides in each triangle are
Notice we have equal ratios and thus a proportion. However, we still must confirm that the included angles are congruent. The measure for this angle is not given in triangle I, but we can calculate since all three angles must add up to 180 degrees. Calculation tells us that the measure is 98 degrees, which unfortunately does not equal the 110 from triangle II. Therefore, we have no SAS and therefore no similarity between I and II.
Transitioning to I and III, we only have angles in triangle III, so we are unable to use either SSS or SAS. However, we previously calculated the measure third angle in triangle I to be 98. Therefore, two of our angles are congruent, meaning we have AA and thus similarity.
Regarding II and III, we can use some logic. Since we know I and III are similar, then if II and III were also similar, then we could use the transitive property to conclude that I and II are also similar. But we know this is false, so II and III cannot be similar.
Therefore, the only two similar triangles are I and III.
We must remember that there are three ways to prove triangles are similar.
- At least two angles in one triangle are congruent to angles in another (AA)
- All three pairs of corresponding sides are proportional (SSS)
- Two pairs of corresponding sides are proportional and the angles between those sides are congruent (SAS)
Comparing triangles I and II, we only have one angle and two sides in trinagle II, so attempting to use either AA or SSS for similarity will not work, leaving SAS as the only option. If we compare the two given sides in each triangle, we notice that the ratio of the longer side in triangle I to the longer side in triangle II is
The ratio of the shorter sides in each triangle are
Notice we have equal ratios and thus a proportion. However, we still must confirm that the included angles are congruent. The measure for this angle is not given in triangle I, but we can calculate since all three angles must add up to 180 degrees. Calculation tells us that the measure is 98 degrees, which unfortunately does not equal the 110 from triangle II. Therefore, we have no SAS and therefore no similarity between I and II.
Transitioning to I and III, we only have angles in triangle III, so we are unable to use either SSS or SAS. However, we previously calculated the measure third angle in triangle I to be 98. Therefore, two of our angles are congruent, meaning we have AA and thus similarity.
Regarding II and III, we can use some logic. Since we know I and III are similar, then if II and III were also similar, then we could use the transitive property to conclude that I and II are also similar. But we know this is false, so II and III cannot be similar.
Therefore, the only two similar triangles are I and III.
Are these triangles similar? If so, state the scale factor.
Are these triangles similar? If so, state the scale factor.
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For both triangles, we are given the "legs." Based on their relative lenghts, we can see that 2 corresponds with 3, and 7 corresponds with 10.5. First we need to make sure that these two triangles are similar. We can do this by comparing the ratios of corresponding sides:
There are a couple of ways to go from here. One would be to cross-multiply:
the ratios are equal, so the triangles are similar, and the scale factor is 
.
For both triangles, we are given the "legs." Based on their relative lenghts, we can see that 2 corresponds with 3, and 7 corresponds with 10.5. First we need to make sure that these two triangles are similar. We can do this by comparing the ratios of corresponding sides:
There are a couple of ways to go from here. One would be to cross-multiply:
Are these triangles similar? If so, state the scale factor.
Are these triangles similar? If so, state the scale factor.
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Based on their positions relative to the congruent angles, and their relative lengths, we can see that 1.5 corresponds to 6, and 8 corresponds to 30. If the ratios of corresponding sides are equal, then the triangles are congruent:
We can compare these in a couple different ways. One would be to cross-multiply:
These triangles are not similar.
Based on their positions relative to the congruent angles, and their relative lengths, we can see that 1.5 corresponds to 6, and 8 corresponds to 30. If the ratios of corresponding sides are equal, then the triangles are congruent:
We can compare these in a couple different ways. One would be to cross-multiply:
These triangles are not similar.
Which of the following shifts are incorrect?
Which of the following shifts are incorrect?
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The actual shift for 
is 
.
The actual shift for
Corresponding sides of two triangles have measures of 
and 
. If another side of the small triangle is 
, what is the value of the bigger triangles corresponding side?
Corresponding sides of two triangles have measures of
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We can find the scale factor by dividing 18 by 6.
We get that the scale factor is 3.
Multiplying the other side by 3, we get the new side, 21.
We can find the scale factor by dividing 18 by 6.
We get that the scale factor is 3.
Multiplying the other side by 3, we get the new side, 21.
A right triangle has side lengths of 
, 
, and 
. A similar right triangle has sides of 
, 
, and 
. What is 
?
A right triangle has side lengths of
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Similar triangles by defnition have proportional sides. We can divide corresponding parts in this case to find the scale factor.
Corresponding parts are the two smallest sides, the medium sides, and the largest sides.
Thus:
is the scale factor.
Then, we use this to find the missing side. 
Therefore, 
.
Similar triangles by defnition have proportional sides. We can divide corresponding parts in this case to find the scale factor.
Corresponding parts are the two smallest sides, the medium sides, and the largest sides.
Thus:
Then, we use this to find the missing side.
Therefore,
Which set of the following triangle dimensions does NOT have the same proportions as a 3-4-5 triangle?
Which set of the following triangle dimensions does NOT have the same proportions as a 3-4-5 triangle?
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In order to determine whether if the dimensions of the triangle are of the same proportions, the ratios of the dimensions must also be the same as the 3-4-5 triangle.
The following scale factors multiplied to the 3-4-5 triangle yield similar proportions.
The only dimensions that cannot be attained by multiplying a particular scale factor with the 3-4-5 is:
In order to determine whether if the dimensions of the triangle are of the same proportions, the ratios of the dimensions must also be the same as the 3-4-5 triangle.
The following scale factors multiplied to the 3-4-5 triangle yield similar proportions.
The only dimensions that cannot be attained by multiplying a particular scale factor with the 3-4-5 is:
These triangles are similar. Use this to solve for x:
These triangles are similar. Use this to solve for x:
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Initially we are given the hypotenuse and one leg of the large triangle, but both legs of the small triangle. To set up a proportion, we need to know both legs of the large triangle, and we can solve for the missing one using the pythagorean theorem:
, where a and b are the legs and c is the hypotenuse.
subtract 64 from both sides
take the square root of both sides
Now that we have both legs, we can see that 15 corresponds with x, and 8 corresponds with 6, so we can set up a proportion to solve for x:
divide both sides by 8
Initially we are given the hypotenuse and one leg of the large triangle, but both legs of the small triangle. To set up a proportion, we need to know both legs of the large triangle, and we can solve for the missing one using the pythagorean theorem:
Now that we have both legs, we can see that 15 corresponds with x, and 8 corresponds with 6, so we can set up a proportion to solve for x:
These two triangles are similar. Solve for x:
These two triangles are similar. Solve for x:
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Before we can set up a proportion, we need to know the third side of the larger triangle, since that side is actually the one that corresponds to 6, since both are the shortest sides in their respecitve triangles. We can solve for that side using the Pythagorean Theorem, 
subtract 100 from both sides
take the square root
Now we can set up a proportion comparing corresponding sides to solve for x:
cross-multiply
divide by 7.5
Before we can set up a proportion, we need to know the third side of the larger triangle, since that side is actually the one that corresponds to 6, since both are the shortest sides in their respecitve triangles. We can solve for that side using the Pythagorean Theorem,
Now we can set up a proportion comparing corresponding sides to solve for x:
Which of the following is not a theorem to prove that triangles are similar?
Which of the following is not a theorem to prove that triangles are similar?
Tap to see back →
ASA (Angle Side Angle) is a theorem to prove triangle congruency.
In this case, we only need two angles to prove that two triangles are similar, so the last side in ASA is unnecessary for this question.
For this purpose, we use the theorem AA instead.
ASA (Angle Side Angle) is a theorem to prove triangle congruency.
In this case, we only need two angles to prove that two triangles are similar, so the last side in ASA is unnecessary for this question.
For this purpose, we use the theorem AA instead.
What does the scale factor of a dilation need to be to ensure that triangles are not only similar but also congruent?
What does the scale factor of a dilation need to be to ensure that triangles are not only similar but also congruent?
Tap to see back →
The scale factor of a dilation tells us what we multiply corresponding sides by to get the new side lengths. In this case, we want these lengths to be the same to get congruent triangles. Thus, we must be looking for the multiplicative identity, which is 1.
The scale factor of a dilation tells us what we multiply corresponding sides by to get the new side lengths. In this case, we want these lengths to be the same to get congruent triangles. Thus, we must be looking for the multiplicative identity, which is 1.
One triangle has side measures 2, 4, and 5. Another has sides 4, 8, and 10. Are these triangles similar?
One triangle has side measures 2, 4, and 5. Another has sides 4, 8, and 10. Are these triangles similar?
Tap to see back →
To determine if the triangles are similar, set up a proportion.
2/4 = 4/8 = 5/10
When we do this, we cross multiply to get a true statement.
Or, we can find the scale factor.
Since the scale factor is 2 for all three lengths, it becomes clear that these triangles are similar.
To determine if the triangles are similar, set up a proportion.
2/4 = 4/8 = 5/10
When we do this, we cross multiply to get a true statement.
Or, we can find the scale factor.
Since the scale factor is 2 for all three lengths, it becomes clear that these triangles are similar.
One triangle has side measures , , and . Another has side lengths , , and . Are these triangles similar?
One triangle has side measures
Tap to see back →
Two triangles are similar if and only if their side lengths are proportional.
In this case, two of the sides are proportional, leading us to a scale factor of 2.
However, with the last side, which is not our side length.
Thus, these pair of sides are not proportional and therefore our triangles cannot be similar.
Two triangles are similar if and only if their side lengths are proportional.
In this case, two of the sides are proportional, leading us to a scale factor of 2.
However, with the last side,
Thus, these pair of sides are not proportional and therefore our triangles cannot be similar.
Which of the following triangles are similar?
Which of the following triangles are similar?
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We must remember that there are three ways to prove triangles are similar.
- At least two angles in one triangle are congruent to angles in another (AA)
- All three pairs of corresponding sides are proportional (SSS)
- Two pairs of corresponding sides are proportional and the angles between those sides are congruent (SAS)
Comparing triangles I and II, we only have one angle and two sides in trinagle II, so attempting to use either AA or SSS for similarity will not work, leaving SAS as the only option. If we compare the two given sides in each triangle, we notice that the ratio of the longer side in triangle I to the longer side in triangle II is
The ratio of the shorter sides in each triangle are
Notice we have equal ratios and thus a proportion. However, we still must confirm that the included angles are congruent. The measure for this angle is not given in triangle I, but we can calculate since all three angles must add up to 180 degrees. Calculation tells us that the measure is 98 degrees, which unfortunately does not equal the 110 from triangle II. Therefore, we have no SAS and therefore no similarity between I and II.
Transitioning to I and III, we only have angles in triangle III, so we are unable to use either SSS or SAS. However, we previously calculated the measure third angle in triangle I to be 98. Therefore, two of our angles are congruent, meaning we have AA and thus similarity.
Regarding II and III, we can use some logic. Since we know I and III are similar, then if II and III were also similar, then we could use the transitive property to conclude that I and II are also similar. But we know this is false, so II and III cannot be similar.
Therefore, the only two similar triangles are I and III.
We must remember that there are three ways to prove triangles are similar.
- At least two angles in one triangle are congruent to angles in another (AA)
- All three pairs of corresponding sides are proportional (SSS)
- Two pairs of corresponding sides are proportional and the angles between those sides are congruent (SAS)
Comparing triangles I and II, we only have one angle and two sides in trinagle II, so attempting to use either AA or SSS for similarity will not work, leaving SAS as the only option. If we compare the two given sides in each triangle, we notice that the ratio of the longer side in triangle I to the longer side in triangle II is
The ratio of the shorter sides in each triangle are
Notice we have equal ratios and thus a proportion. However, we still must confirm that the included angles are congruent. The measure for this angle is not given in triangle I, but we can calculate since all three angles must add up to 180 degrees. Calculation tells us that the measure is 98 degrees, which unfortunately does not equal the 110 from triangle II. Therefore, we have no SAS and therefore no similarity between I and II.
Transitioning to I and III, we only have angles in triangle III, so we are unable to use either SSS or SAS. However, we previously calculated the measure third angle in triangle I to be 98. Therefore, two of our angles are congruent, meaning we have AA and thus similarity.
Regarding II and III, we can use some logic. Since we know I and III are similar, then if II and III were also similar, then we could use the transitive property to conclude that I and II are also similar. But we know this is false, so II and III cannot be similar.
Therefore, the only two similar triangles are I and III.
Are these triangles similar? If so, state the scale factor.
Are these triangles similar? If so, state the scale factor.
Tap to see back →
For both triangles, we are given the "legs." Based on their relative lenghts, we can see that 2 corresponds with 3, and 7 corresponds with 10.5. First we need to make sure that these two triangles are similar. We can do this by comparing the ratios of corresponding sides:
There are a couple of ways to go from here. One would be to cross-multiply:
the ratios are equal, so the triangles are similar, and the scale factor is .
For both triangles, we are given the "legs." Based on their relative lenghts, we can see that 2 corresponds with 3, and 7 corresponds with 10.5. First we need to make sure that these two triangles are similar. We can do this by comparing the ratios of corresponding sides:
There are a couple of ways to go from here. One would be to cross-multiply:
Are these triangles similar? If so, state the scale factor.
Are these triangles similar? If so, state the scale factor.
Tap to see back →
Based on their positions relative to the congruent angles, and their relative lengths, we can see that 1.5 corresponds to 6, and 8 corresponds to 30. If the ratios of corresponding sides are equal, then the triangles are congruent:
We can compare these in a couple different ways. One would be to cross-multiply:
These triangles are not similar.
Based on their positions relative to the congruent angles, and their relative lengths, we can see that 1.5 corresponds to 6, and 8 corresponds to 30. If the ratios of corresponding sides are equal, then the triangles are congruent:
We can compare these in a couple different ways. One would be to cross-multiply:
These triangles are not similar.