How to find if right triangles are similar - Math
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A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
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The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
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You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
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Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
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In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
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Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
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; 
is a right angle; 
; 
; 
Find 
.
Find
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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, 
.
Since
Because the two are similar triangles,
Therefore,
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Which of the following is sufficient to say that two right triangles are similar?
Which of the following is sufficient to say that two right triangles are similar?
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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.
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.
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Which of the following statements is true regarding the two triangles?
Which of the following statements is true regarding the two triangles?
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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.
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.
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Two triangles, 
and 
, are similar when:
Two triangles,
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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.
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.
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and 
are triangles.
Are 
and 
similar?
Are
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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.
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
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.
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and 
are similar triangles.
What is the length of 
?
What is the length of
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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 
.
Since
Substitute the known values.
Cross-multiply and simplify.
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Are these triangles similar? If so, list the scale factor.
Are these triangles similar? If so, list the scale factor.
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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.
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.
Now we can compare all three ratios of corresponding sides:
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
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Are these right triangles similar? If so, state the scale factor.
Are these right triangles similar? If so, state the scale factor.
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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.
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:
The other triangle is missing one of the legs rather than the hypotenuse, so we'll adjust accordingly:
Now we can compare ratios of corresponding sides:
The first ratio simplifies to
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Are these triangles similar? Give a justification.
Are these triangles similar? Give a justification.
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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.
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
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Refer to the above figure.
True, false, or inconclusive: 
.
Refer to the above figure.
True, false, or inconclusive:
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is an altitude of 
, so it divides the triangle into two smaller triangles similar to each other - that is, if we match the shorter legs, the longer legs, and the hypotenuses, the similarity statement is
.
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Refer to the above diagram.
True or false: 
Refer to the above diagram.
True or false:
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The distance from the origin to 
is the absolute value of the 
-coordinate of 
, which is 
. Similarly, 
, 
, and 
. Also, since the axes intersect at right angles, 
and 
are both right, and, consequently, congruent.
According to the Side-Angle-Side Similarity Theorem (SASS), if two sides of a triangle are in proportion to the corresponding sides of a second triangle, and their included angles are congruent, the triangles are similar.
We can test the proportion statement
by substituting:
Test the truth of this statement by comparing their cross products:
The cross-products are equal, making the proportion statement true, so two pairs of sides are in proportion. Also, their included angles 
and 
are congruent. This sets up the conditions of SASS, so 
.
The distance from the origin to
According to the Side-Angle-Side Similarity Theorem (SASS), if two sides of a triangle are in proportion to the corresponding sides of a second triangle, and their included angles are congruent, the triangles are similar.
We can test the proportion statement
by substituting:
Test the truth of this statement by comparing their cross products:
The cross-products are equal, making the proportion statement true, so two pairs of sides are in proportion. Also, their included angles
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Given: 
and 
.
and 
are both right angles.
True or false: From the given information, it follows that 
.
Given:
True or false: From the given information, it follows that
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If we seek to prove that 
, then 
, 
, and 
correspond to 
, 
, and 
, respectively.
By the Side-Angle-Side Similarity Theorem (SASS), if two sides of a triangle are in proportion with the corresponding sides of another triangle, and the included angles are congruent, then the triangles are similar.
and 
, so by the Division Property of Equality, 
. Also, 
and 
, their respective included angles, are both right angles, so 
. The conditions of SASS are met, so 
If we seek to prove that
By the Side-Angle-Side Similarity Theorem (SASS), if two sides of a triangle are in proportion with the corresponding sides of another triangle, and the included angles are congruent, then the triangles are similar.
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A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
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The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
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You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
Tap to reveal answer
Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
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In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
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Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.
Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.
We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.
We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:
_a_2 + _b_2 = _c_2
152 + 202 = _c_2
625 = _c_2
c = 25
The length of BD is 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths of AB, BC, CD, and DA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
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A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
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The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
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You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
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Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
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