Special triangles - HiSET
Card 1 of 12
Two of a triangle's interior angles measure 
and 
, respectively. If this triangle's hypotenuse is 
long, what are the lengths of its other sides?
Two of a triangle's interior angles measure 
Tap to reveal answer
A triangle that has interior angles of 
and 
is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be 
because a triangle's interior angles always sum to 
, allowing us to solve for the third angle like so:
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of 
. The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is 
. So, we need to set 
equivalent to 
and solve for 
.
As you can see, for this particular triangle, 
. Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the 
angle will be equal to 
inches; since 
, this side's length is 
. The side opposite the 
angle will be equal to 
. Substituting in 
into this expression, we find that this side has a length of 
.
Thus, the correct answer is 
.
A triangle that has interior angles of 
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of 
As you can see, for this particular triangle, 
Thus, the correct answer is 
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Examine the above triangle. Which of the following correctly gives the area of 
?
Examine the above triangle. Which of the following correctly gives the area of 
Tap to reveal answer
Since 
is a right angle - that is, 
- and 
, it follows that
,
making 
a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
,
the correct response.
Since 
making 
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
the correct response.
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Examine the above triangle. Which of the following correctly gives the perimeter of 
?
Examine the above triangle. Which of the following correctly gives the perimeter of 
Tap to reveal answer
Since 
is a right angle - that is, 
- and 
, it follows that
,
making 
a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
.
Since 
making 
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
← Didn't Know|Knew It →
Two of a triangle's interior angles measure and , respectively. If this triangle's hypotenuse is long, what are the lengths of its other sides?
Two of a triangle's interior angles measure
Tap to reveal answer
A triangle that has interior angles of and is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be because a triangle's interior angles always sum to , allowing us to solve for the third angle like so:
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of . The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is . So, we need to set equivalent to and solve for .
As you can see, for this particular triangle, . Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the angle will be equal to inches; since , this side's length is . The side opposite the angle will be equal to . Substituting in into this expression, we find that this side has a length of .
Thus, the correct answer is .
A triangle that has interior angles of
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of
As you can see, for this particular triangle,
Thus, the correct answer is
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Examine the above triangle. Which of the following correctly gives the area of ?
Examine the above triangle. Which of the following correctly gives the area of
Tap to reveal answer
Since is a right angle - that is, - and , it follows that
,
making a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
,
the correct response.
Since
making
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
the correct response.
← Didn't Know|Knew It →
Examine the above triangle. Which of the following correctly gives the perimeter of ?
Examine the above triangle. Which of the following correctly gives the perimeter of
Tap to reveal answer
Since is a right angle - that is, - and , it follows that
,
making a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
.
Since
making
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
← Didn't Know|Knew It →
Two of a triangle's interior angles measure and , respectively. If this triangle's hypotenuse is long, what are the lengths of its other sides?
Two of a triangle's interior angles measure
Tap to reveal answer
A triangle that has interior angles of and is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be because a triangle's interior angles always sum to , allowing us to solve for the third angle like so:
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of . The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is . So, we need to set equivalent to and solve for .
As you can see, for this particular triangle, . Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the angle will be equal to inches; since , this side's length is . The side opposite the angle will be equal to . Substituting in into this expression, we find that this side has a length of .
Thus, the correct answer is .
A triangle that has interior angles of
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of
As you can see, for this particular triangle,
Thus, the correct answer is
← Didn't Know|Knew It →
Examine the above triangle. Which of the following correctly gives the area of ?
Examine the above triangle. Which of the following correctly gives the area of
Tap to reveal answer
Since is a right angle - that is, - and , it follows that
,
making a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
,
the correct response.
Since
making
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
the correct response.
← Didn't Know|Knew It →
Examine the above triangle. Which of the following correctly gives the perimeter of ?
Examine the above triangle. Which of the following correctly gives the perimeter of
Tap to reveal answer
Since is a right angle - that is, - and , it follows that
,
making a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
.
Since
making
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
← Didn't Know|Knew It →
Two of a triangle's interior angles measure and , respectively. If this triangle's hypotenuse is long, what are the lengths of its other sides?
Two of a triangle's interior angles measure
Tap to reveal answer
A triangle that has interior angles of and is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be because a triangle's interior angles always sum to , allowing us to solve for the third angle like so:
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of . The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is . So, we need to set equivalent to and solve for .
As you can see, for this particular triangle, . Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the angle will be equal to inches; since , this side's length is . The side opposite the angle will be equal to . Substituting in into this expression, we find that this side has a length of .
Thus, the correct answer is .
A triangle that has interior angles of
Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.
We're told that the hypotenuse of our triangle has a length of
As you can see, for this particular triangle,
Thus, the correct answer is
← Didn't Know|Knew It →
Examine the above triangle. Which of the following correctly gives the area of ?
Examine the above triangle. Which of the following correctly gives the area of
Tap to reveal answer
Since is a right angle - that is, - and , it follows that
,
making a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
,
the correct response.
Since
making
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The area of a right triangle is equal to half the product of the lengths of its legs, so
the correct response.
← Didn't Know|Knew It →
Examine the above triangle. Which of the following correctly gives the perimeter of ?
Examine the above triangle. Which of the following correctly gives the perimeter of
Tap to reveal answer
Since is a right angle - that is, - and , it follows that
,
making a 30-60-90 triangle.
By the 30-60-90 Triangle Theorem,
,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
.
Since
making
By the 30-60-90 Triangle Theorem,
and
Refer to the diagram below:
The perimeter - the sum of the sidelengths - is
← Didn't Know|Knew It →