### All Trigonometry Resources

## Example Questions

### Example Question #1 : Identifying Similar Triangles

What does the scale factor of a dilation need to be to ensure that triangles are not only similar but also congruent?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #1 : Identifying Similar Triangles

Which of the following is not a theorem to prove that triangles are similar?

**Possible Answers:**

AA

ASA

SAS

SSS

HL

**Correct answer:**

ASA

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.

### Example Question #1 : Similar Triangles

One triangle has side measures , , and . Another has side lengths , , and . Are these triangles similar?

**Possible Answers:**

There is not enough information given.

Those can't be the side lengths of triangles.

Triangles can't be similar!

No

Yes

**Correct answer:**

No

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.

### Example Question #2 : Similar Triangles

Which of the following triangles are similar?

**Possible Answers:**

I and II

II and III

I and III

I, II, and III

None of the triangles are similar

**Correct answer:**

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.

### Example Question #1 : Identifying Similar Triangles

One triangle has side measures , , and . Another has sides , , and . Are these triangles similar?

**Possible Answers:**

They are congruent triangles

No

Yes

There is not enough information

**Correct answer:**

Yes

To determine if the triangles are similar, set up a proportion.

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.

### Example Question #1 : Identifying Similar Triangles

Are these triangles similar? If so, state the scale factor.

**Possible Answers:**

Yes - scale factor

Yes - scale factor

No

Yes - scale factor

Yes - scale factor

**Correct answer:**

Yes - scale factor

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 .

### Example Question #2 : Identifying Similar Triangles

Are these triangles similar? If so, state the scale factor.

**Possible Answers:**

Yes - scale factor

Yes - scale factor

Yes - scale factor

No, they are not similar

Not enough information

**Correct answer:**

No, they 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.

### Example Question #1 : Similar Triangles

Which of the following shifts are incorrect?

**Possible Answers:**

**Correct answer:**

The actual shift for is .

### Example Question #1 : Proportions In Similar Triangles

A right triangle has side lengths of , , and . A similar right triangle has sides of , , and . What is ?

**Possible Answers:**

There is not enough information to determine.

**Correct answer:**

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, .

### Example Question #6 : Similar Triangles

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?

**Possible Answers:**

**Correct answer:**

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.

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