Basic Geometry : Right Triangles

Example Questions

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Example Question #1 : Right Triangles

is a right angle; ; ;

Find .

Explanation:

Since  and  is a right angle,   is also a right angle.

is the hypotenuse of the first triangle; since one of its legs  is half the length of that hypotenuse,  is 30-60-90 with  the shorter leg and  the longer.

Because the two are similar triangles,  is the hypotenuse of the second triangle, and  is its longer leg.

Therefore, .

Example Question #2 : Right Triangles

Which of the following is sufficient to say that two right triangles are similar?

Two sides and one angle are congruent.

All the angles are congruent.

Two angles and one side are congruent.

Two of the sides are the same.

All the angles are congruent.

Explanation:

If all three angles of a triangle are congruent but the sides are not, then one of the triangles is a scaled up version of the other. When this happens the proportions between the sides still remains unchanged which is the criteria for similarity.

Example Question #3 : Right Triangles

Which of the following statements is true regarding the two triangles?

The triangles are not similar because they are not the same size.

The triangles are similar by Side-Angle-Side

The triangles are similar because they are both right triangles.

We do not have enough information to show the triangles are similar.

The triangles are similar by Hypotenuse-Leg

The triangles are similar by Side-Angle-Side

Explanation:

Though we must do a little work, we can show these triangles are similar. First, right triangles are not necessarily always similar. They must meet the necessary criteria like any other triangles; furthermore, there is no Hypotenuse-Leg Theorem for similarity, only for congruence; therefore, we can eliminate two answer choices.

However, we can use the Pythagorean Theorem with the smaller triangle to find the missing leg. Doing so gives us a length of 48. Comparing the ratio of the shorter legs in each trangle to the ratio of the longer legs we get

In both cases, the leg of the larger triangle is twice as long as the corresponding leg in the smaller triangle. Given that the angle between the two legs is a right angle in each triangle, these angles are congruent. We now have enough evidence to conclude similarity by Side-Angle-Side.

Example Question #4 : Right Triangles

Two triangles,  and , are similar when:

Their corresponding angles are equal AND their corresponding lengths are equal.

Their corresponding angles are equal.

Their corresponding lengths are proportional.

Their corresponding angles are equal AND their corresponding lengths are proportional.

Their corresponding angles are equal AND their corresponding lengths are proportional.

Explanation:

The Similar Figures Theorem holds that similar figures have both equal corresponding angles and proportional corresponding lengths. Either condition alone is not sufficient.  If two figures have both equal corresponding angles and equal corresponding lengths then they are congruent, not similar.

Example Question #5 : Right Triangles

and  are triangles.

Are  and  similar?

There is not enough information given to answer this question.

Yes, because  and  are both right triangles.

Yes, because  and  look similar.

No, because  and  are not the same size.

There is not enough information given to answer this question.

Explanation:

The Similar Figures Theorem holds that similar figures have both equal corresponding angles and proportional corresponding lengths. In other words, we need to know both the measures of the corresponding angles and the lengths of the corresponding sides. In this case, we know only the measures of  and . We don't know the measures of any of the other angles or the lengths of any of the sides, so we cannot answer the question -- they might be similar, or they might not be.

It's not enough to know that both figures are right triangles, nor can we assume that angles are the same measurement because they appear to be.

Similar triangles do not have to be the same size.

Example Question #6 : Right Triangles

and  are similar triangles.

What is the length of ?

Explanation:

Since  and  are similar triangles, we know that they have proportional corresponding lengths.  We must determine which sides correspond.  Here, we know  corresponds to  because both line segments lie opposite  angles and between  and  angles. Likewise, we know corresponds to  because both line segments lie opposite  angles and between  and  angles.  We can use this information to set up a proportion and solve for the length of .

Substitute the known values.

Cross-multiply and simplify.

and  result from setting up an incorrect proportion.  results from incorrectly multiplying  and .

Example Question #7 : Right Triangles

Are these triangles similar? Give a justification.

Not enough information - we would need to know at least one side length in each triangle

No - the side lengths are not proportional

Yes - they LOOK like they're similar

Yes - the triangles are similar by AA

No - the angles are not the same

Yes - the triangles are similar by AA

Explanation:

These triangles were purposely drawn misleadingly. Just from glancing at them, the angles that appear to correspond are given different angle measures, so they don't "look" similar. However, if we subtract, we figure out that the missing angle in the triangle with the 66-degree angle must be 24 degrees, since . Similarly, the missing angle in the triangle with the 24-degree angle must be 66 degress. This means that all 3 corresponding pairs of angles are congruent, making the triangles similar.

Example Question #8 : Right Triangles

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

Yes-scale factor

Cannot be determined - we need to know all three sides of both triangles

Yes - scale factor

No

Yes - scale factor

Yes - scale factor

Explanation:

The two triangles are similar, but we can't be sure of that until we can compare all three corresponding pairs of sides and make sure the ratios are the same. In order to do that, we first have to solve for the missing sides using the Pythagorean Theorem.

The smaller triangle is missing not the hypotenuse, c, but one of the legs, so we'll use the formula slightly differently.

subtract 36 from both sides

Now we can compare all three ratios of corresponding sides:

one way of comparing these ratios is to simplify them.

We can simplify the leftmost ratio by dividing top and bottom by 3 and getting

We can simplify the middle ratio by dividing top and bottom by 4 and getting .

Finally, we can simplify the ratio on the right by dividing top and bottom by 5 and getting .

This means that the triangles are definitely similar, and is the scale factor.

Example Question #9 : Right Triangles

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

Not enough information to be determined

Yes - scale factor

Yes - scale factor

Yes - scale factor

No - the side lengths are not proportional

No - the side lengths are not proportional

Explanation:

In order to compare these triangles and determine if they are similar, we need to know all three side lengths in both triangles. To get the missing ones, we can use Pythagorean Theorem:

take the square root

The other triangle is missing one of the legs rather than the hypotenuse, so we'll adjust accordingly:

subtract 36 from both sides

Now we can compare ratios of corresponding sides:

The first ratio simplifies to , but we can't simplify the others any more than they already are. The three ratios clearly do not match, so these are not similar triangles.

Example Question #10 : Right Triangles

Given:  and .

and  are both right angles.

True or false: From the given information, it follows that .

True

False