Triangles - Math
Card 0 of 4048
is an equilateral triangle. Points 
are the midpoints of 
, respectively. 
is constructed.
Which is the greater quantity?
(a) The area of 
(b) Twice the area of 
Which is the greater quantity?
(a) The area of
(b) Twice the area of
If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then four triangles, congruent to each other and similar to the larger triangle, are formed. Therefore, one of these triangles - specifically, 
- would have one-fourth the area of 
. This means 
has more than twice the area of 
.
Note that the fact that the triangle is equilateral is irrelevant.
If segments are constructed in which the endpoints form the midpoints of the sides of a triangle, then four triangles, congruent to each other and similar to the larger triangle, are formed. Therefore, one of these triangles - specifically,
Note that the fact that the triangle is equilateral is irrelevant.
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Which of the following could be the three sidelengths of an equilateral triangle?
Which of the following could be the three sidelengths of an equilateral triangle?
By definition, an equilateral triangle has three sides of equal length. We can identify the equilateral triangle by converting the given sidelengths to the same units and comparing them.
We can eliminate the following by showing that at least two sidelengths differ.
2 yards = 
feet.
Two sides have lengths 6 feet and 7 feet, so we can eliminate this choice.
4 feet = 
inches
Two sides have lengths 48 inches and 50 inches, so we can eliminate this choice.
5 feet = 
inches
Two sides have lengths 48 inches and 60 inches, so we can eliminate this choice.
yards = 
feet
Two sides have lengths 4 feet and 5 feet, so we can eliminate this choice.
yards = 
feet = 
inches
All three sides have the same length, making this the triangle equilateral. This choice is correct.
By definition, an equilateral triangle has three sides of equal length. We can identify the equilateral triangle by converting the given sidelengths to the same units and comparing them.
We can eliminate the following by showing that at least two sidelengths differ.
2 yards =
Two sides have lengths 6 feet and 7 feet, so we can eliminate this choice.
4 feet =
Two sides have lengths 48 inches and 50 inches, so we can eliminate this choice.
5 feet =
Two sides have lengths 48 inches and 60 inches, so we can eliminate this choice.
Two sides have lengths 4 feet and 5 feet, so we can eliminate this choice.
All three sides have the same length, making this the triangle equilateral. This choice is correct.
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What is the hypotenuse of a right triangle with sides 5 and 8?
What is the hypotenuse of a right triangle with sides 5 and 8?
Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
_a_2 + _b_2 = _c_2
52 + 82 = _c_2
25 + 64 = _c_2
89 = _c_2
c = √89
Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
_a_2 + _b_2 = _c_2
52 + 82 = _c_2
25 + 64 = _c_2
89 = _c_2
c = √89
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Which is the greater quantity?
(a) The hypotenuse of a right triangle with legs 
and 
.
(b) The hypotenuse of a right triangle with legs 
and 
.
Which is the greater quantity?
(a) The hypotenuse of a right triangle with legs
(b) The hypotenuse of a right triangle with legs
The hypotenuses of the triangles measure as follows:
(a) 
(b) 
, so 
, making (a) the greater quantity.
The hypotenuses of the triangles measure as follows:
(a)
(b)
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Which is the greater quantity?
(a) The hypotenuse of a 
right triangle with a leg of length 20
(b) The hypotenuse of a right triangle with legs of length 19 and 21
Which is the greater quantity?
(a) The hypotenuse of a
(b) The hypotenuse of a right triangle with legs of length 19 and 21
The hypotenuses of the triangles measure as follows:
(a) 
(b) 
, so 
, making (b) the greater quantity
The hypotenuses of the triangles measure as follows:
(a)
(b)
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A right triangle has a leg 
feet long and a hypotenuse 
feet long. Which is the greater quantity?
(a) The length of the second leg of the triangle
(b) 60 inches
A right triangle has a leg
(a) The length of the second leg of the triangle
(b) 60 inches
The length of the second leg can be calculated using the Pythagorean Theorem. Set 
:
The second leg therefore measures 
inches.
The length of the second leg can be calculated using the Pythagorean Theorem. Set
The second leg therefore measures
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What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?
What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?
Since we're dealing with right triangles, we can use the Pythagorean Theorem (
). In this formula, a and b are the sides, while c is the hypotenuse. The hypotenuse of a right triangle is the longest side and the side that is opposite the right angle. Now, we can plug into our formula, which looks like this: 
We simplify and get 
. At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.
Since we're dealing with right triangles, we can use the Pythagorean Theorem (
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The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?
(A) The length of one side of the pentagon
(B) One and one-half feet
The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?
(A) The length of one side of the pentagon
(B) One and one-half feet
By the Pythagorean Theorem, the hypotenuse of the right triangle is
inches, making its perimeter
inches.
The pentagon in question has sides of length 75% of 112, or
.
Since a pentagon has five sides of equal length, each side will have measure
inches.
One and a half feet are equivalent to 
inches, so (B) is the greater quantity.
By the Pythagorean Theorem, the hypotenuse of the right triangle is
The pentagon in question has sides of length 75% of 112, or
Since a pentagon has five sides of equal length, each side will have measure
One and a half feet are equivalent to
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The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.
Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.
Which is the greater quantity?
(A) The distance Cary runs
(B) One-fourth of a mile
The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.
Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.
Which is the greater quantity?
(A) The distance Cary runs
(B) One-fourth of a mile
By the Pythagorean Theorem, the distance from B to C is
feet
Cary runs
feet
Since 5,280 feet make a mile, one-fourth of a mile is equal to
feet.
(B) is greater
By the Pythagorean Theorem, the distance from B to C is
Cary runs
Since 5,280 feet make a mile, one-fourth of a mile is equal to
(B) is greater
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Give the length of the hypotenuse of the above right triangle in terms of 
.
Give the length of the hypotenuse of the above right triangle in terms of
If we let 
be the length of the hypotenuse, then by the Pythagorean theorem,
If we let
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In Square 
. 
is the midpoint of 
, 
is the midpoint of 
, and 
is the midpoint of 
. Construct the line segments 
and 
.
Which is the greater quantity?
(a) 
(b) 
In Square
Which is the greater quantity?
(a)
(b)
The figure referenced is below:
For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.
and 
are midpoints of their respective sides, so 
, making 
the hypotenuse of a triangle with legs of length 2 and 2. Therefore,
.
Also, 
, and since 
is the midpoint of 
, 
. 
, making 
the hypotenuse of a triangle with legs of length 1 and 4. Therefore,
, so 
The figure referenced is below:
For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.
Also,
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Figure NOT drawn to scale.
In the above figure, 
is a right angle.
What is the length of 
?
Figure NOT drawn to scale.
In the above figure,
What is the length of
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
.
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
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Figure NOT drawn to scale.
In the above figure, 
is a right angle.
What is the length of 
?
Figure NOT drawn to scale.
In the above figure,
What is the length of
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
.
Their corresponding sides are in proportion, so, setting the ratios of the long legs to the short legs equal to each other,
By the Pythagorean Theorem.
The proportion statement becomes
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the long legs to the short legs equal to each other,
By the Pythagorean Theorem.
The proportion statement becomes
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Given: 
with 
, 
, 
.
Which is the greater quantity?
(a) 
(b) 
Given:
Which is the greater quantity?
(a)
(b)
The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:
; it follows that 
is obtuse, and has measure greater than 
The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:
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Figure NOT drawn to scale.
In the above figure, is a right angle.
What is the length of ?
Figure NOT drawn to scale.
In the above figure,
What is the length of
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
.
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
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Refer to the above right triangle. Which of the following is equal to 
?
Refer to the above right triangle. Which of the following is equal to
By the Pythagorean Theorem,
By the Pythagorean Theorem,
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Given 
with right angle 
, 
Which is the greater quantity?
(a) 
(b) 
Given
Which is the greater quantity?
(a)
(b)
The sum of the measures of the angles of a triangle is 
, so:
This is a 
triangle, so its legs 
and 
are congruent. The quantities are equal.
The sum of the measures of the angles of a triangle is , so:
This is a
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Give the length of one leg of an isosceles right triangle whose area is the same as the right triangle in the above diagram.
Give the length of one leg of an isosceles right triangle whose area is the same as the right triangle in the above diagram.
The area of a triangle is half the product of its height and its base; in a right triangle, the legs, being perpendicular, can serve as these quantites.
The triangle in the diagram has area
square inches.
An isosceles right triangle has two legs of the same length, which we will call 
. The area of that triangle, which is the same as that of the one in the diagram, is therefore
inches.
The area of a triangle is half the product of its height and its base; in a right triangle, the legs, being perpendicular, can serve as these quantites.
The triangle in the diagram has area
An isosceles right triangle has two legs of the same length, which we will call
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The perimeter of a regular octagon is 20% greater than that of the above right triangle. Which is the greater quantity?
(A) The length of one side of the octagon
(B) 3 yards
The perimeter of a regular octagon is 20% greater than that of the above right triangle. Which is the greater quantity?
(A) The length of one side of the octagon
(B) 3 yards
By the Pythagorean Theorem, the shorter leg has length
feet.
The perimeter of the right triangle is therefore
feet.
The octagon has perimeter 20% greater than this, or
feet.
A regular octagon has eight sides of equal length, so each side of this octagon has length
feet, which is equal to 3 yards. This makes the quantities equal.
By the Pythagorean Theorem, the shorter leg has length
The perimeter of the right triangle is therefore
The octagon has perimeter 20% greater than this, or
A regular octagon has eight sides of equal length, so each side of this octagon has length
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The area of a square is equal to that of the above right triangle. Which is the greater quantity?
(A) The sidelength of the square
(B) 4 yards
The area of a square is equal to that of the above right triangle. Which is the greater quantity?
(A) The sidelength of the square
(B) 4 yards
By the Pythagorean Theorem, the shorter leg has length
feet.
The area of a triangle is equal to half the product of its base and height; for a right triangle, the legs can serve as these. The area of the above right triangle is
square feet.
The sidelength is the square root of this; 
, so 
. Therefore each sidelength of the square is just under 11 feet. 4 yards is 12 feet, so (B) is greater.
By the Pythagorean Theorem, the shorter leg has length
The area of a triangle is equal to half the product of its base and height; for a right triangle, the legs can serve as these. The area of the above right triangle is
The sidelength is the square root of this;
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