Triangles - Math
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The base of a right isosceles triangle is 8 inches. The hypotenuse is not the base. What is the area of the triangle in inches?
The base of a right isosceles triangle is 8 inches. The hypotenuse is not the base. What is the area of the triangle in inches?
To find the area of a triangle, multiply the base by the height, then divide by 2. Since the short legs of an isosceles triangle are the same length, we need to know only one to know the other. Since, a short side serves as the base of the triangle, the other short side tells us the height.
To find the area of a triangle, multiply the base by the height, then divide by 2. Since the short legs of an isosceles triangle are the same length, we need to know only one to know the other. Since, a short side serves as the base of the triangle, the other short side tells us the height.
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An isosceles right triangle has a hypotenuse of 
. Find its area.
An isosceles right triangle has a hypotenuse of
In order to calculate the triangle's area, we need to find the lengths of its legs. An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as 
triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.
Now we can calculate the area using the formula
Now, convert to feet.
In order to calculate the triangle's area, we need to find the lengths of its legs. An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as
Now we can calculate the area using the formula
Now, convert to feet.
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In an isosceles right triangle, two sides equal 
. Find the length of side 
.
In an isosceles right triangle, two sides equal
This problem represents the definition of the side lengths of an isosceles right triangle. By definition the sides equal 
, 
, and 
. However, if you did not remember this definition one can also find the length of the side using the Pythagorean theorem 
.
This problem represents the definition of the side lengths of an isosceles right triangle. By definition the sides equal
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ABCD is a square whose side is 
units. Find the length of diagonal AC.
ABCD is a square whose side is
To find the length of the diagonal, given two sides of the square, we can create two equal triangles from the square. The diagonal line splits the right angles of the square in half, creating two triangles with the angles of 
, 
, and 
degrees. This type of triangle is a special right triangle, with the relationship between the side opposite the 
degree angles serving as x, and the side opposite the 
degree angle serving as 
.
Appyling this, if we plug 
in for 
we get that the side opposite the right angle (aka the diagonal) is 
To find the length of the diagonal, given two sides of the square, we can create two equal triangles from the square. The diagonal line splits the right angles of the square in half, creating two triangles with the angles of
Appyling this, if we plug
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The area of a square is 
. Find the length of the diagonal of the square.
The area of a square is
If the area of the square is 
, we know that each side of the square is 
, because the area of a square is 
.
Then, the diagonal creates two 
special right triangles. Knowing that the sides = 
, we can find that the hypotenuse (aka diagonal) is 
If the area of the square is
Then, the diagonal creates two
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What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?
What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?
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An isosceles triangle has a hypotenuse of 
. Find the length of its sides, 
.
An isosceles triangle has a hypotenuse of
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as 
triangles and their side lengths follow a specific pattern that states you can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as
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The measure of the sides of this isosceles right triangle are 
. Find the measure of its hypotenuse, 
.
The measure of the sides of this isosceles right triangle are
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as 
triangles and their side lenghts follow a specific pattern that states you can calculate the length of the hypotenuse of an isoceles triangle by multiplying the length of one of the legs by the square root of 2.
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as
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An isosceles triangle has a base of 6 and a height of 4. What is the perimeter of the triangle?
An isosceles triangle has a base of 6 and a height of 4. What is the perimeter of the triangle?
An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).
The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).
5 + 5 + 6 = 16
An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).
The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).
5 + 5 + 6 = 16
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The hypotenuse of an isosceles right triangle has a measure of 
. Find its perimeter.
The hypotenuse of an isosceles right triangle has a measure of
In order to calculate the triangle's perimeter, we need to find the lengths of its legs. An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as 
triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.
Now we can calculate the perimeter by doubling 
and adding 
.
In order to calculate the triangle's perimeter, we need to find the lengths of its legs. An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as
Now we can calculate the perimeter by doubling
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The side lengths of an isoceles right triangle measure 
. Find its perimeter.
The side lengths of an isoceles right triangle measure
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as 
triangles and their side lenghts follow a specific pattern that states you can calculate the length of the hypotenuse of an isoceles triangle by multiplying the length of one of the legs by the square root of 2.
Now we can calculate the perimeter by doubling 
and adding 
.
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as
Now we can calculate the perimeter by doubling
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A triangle has two angles equal to 
and two sides equal to 
. What is the perimeter of this triangle?
A triangle has two angles equal to
When a triangle has two angles equal to 
, it must be a 
isosceles right triangle.
The pattern for the sides of a 
is 
.
Since two sides are equal to 
, this triangle will have sides of 
.
Add them all together to get 
.
When a triangle has two angles equal to
The pattern for the sides of a
Since two sides are equal to
Add them all together to get
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Acute angles x and y are inside a right triangle. If x is four less than one third of 21, what is y?
Acute angles x and y are inside a right triangle. If x is four less than one third of 21, what is y?
We know that the sum of all the angles must be 180 and we already know one angle is 90, leaving the sum of x and y to be 90.
Solve for x to find y.
One third of 21 is 7. Four less than 7 is 3. So if angle x is 3 then that leaves 87 for angle y.
We know that the sum of all the angles must be 180 and we already know one angle is 90, leaving the sum of x and y to be 90.
Solve for x to find y.
One third of 21 is 7. Four less than 7 is 3. So if angle x is 3 then that leaves 87 for angle y.
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If a right triangle has one leg with a length of 4 and a hypotenuse with a length of 8, what is the measure of the angle between the hypotenuse and its other leg?
If a right triangle has one leg with a length of 4 and a hypotenuse with a length of 8, what is the measure of the angle between the hypotenuse and its other leg?
The first thing to notice is that this is a 30o:60o:90o triangle. If you draw a diagram, it is easier to see that the angle that is asked for corresponds to the side with a length of 4. This will be the smallest angle. The correct answer is 30.
The first thing to notice is that this is a 30o:60o:90o triangle. If you draw a diagram, it is easier to see that the angle that is asked for corresponds to the side with a length of 4. This will be the smallest angle. The correct answer is 30.
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In the figure above, what is the positive difference, in degrees, between the measures of angle ACB and angle CBD?
In the figure above, what is the positive difference, in degrees, between the measures of angle ACB and angle CBD?
In the figure above, angle ADB is a right angle. Because side AC is a straight line, angle CDB must also be a right angle.
Let’s examine triangle ADB. The sum of the measures of the three angles must be 180 degrees, and we know that angle ADB must be 90 degrees, since it is a right angle. We can now set up the following equation.
x + y + 90 = 180
Subtract 90 from both sides.
x + y = 90
Next, we will look at triangle CDB. We know that angle CDB is also 90 degrees, so we will write the following equation:
y – 10 + 2_x_ – 20 + 90 = 180
y + 2_x_ + 60 = 180
Subtract 60 from both sides.
y + 2_x_ = 120
We have a system of equations consisting of x + y = 90 and y + 2_x_ = 120. We can solve this system by solving one equation in terms of x and then substituting this value into the second equation. Let’s solve for y in the equation x + y = 90.
x + y = 90
Subtract x from both sides.
y = 90 – x
Next, we can substitute 90 – x into the equation y + 2_x_ = 120.
(90 – x) + 2_x_ = 120
90 + x = 120
x = 120 – 90 = 30
x = 30
Since y = 90 – x, y = 90 – 30 = 60.
The question ultimately asks us to find the positive difference between the measures of ACB and CBD. The measure of ACB = 2_x_ – 20 = 2(30) – 20 = 40 degrees. The measure of CBD = y – 10 = 60 – 10 = 50 degrees. The positive difference between 50 degrees and 40 degrees is 10.
The answer is 10.
In the figure above, angle ADB is a right angle. Because side AC is a straight line, angle CDB must also be a right angle.
Let’s examine triangle ADB. The sum of the measures of the three angles must be 180 degrees, and we know that angle ADB must be 90 degrees, since it is a right angle. We can now set up the following equation.
x + y + 90 = 180
Subtract 90 from both sides.
x + y = 90
Next, we will look at triangle CDB. We know that angle CDB is also 90 degrees, so we will write the following equation:
y – 10 + 2_x_ – 20 + 90 = 180
y + 2_x_ + 60 = 180
Subtract 60 from both sides.
y + 2_x_ = 120
We have a system of equations consisting of x + y = 90 and y + 2_x_ = 120. We can solve this system by solving one equation in terms of x and then substituting this value into the second equation. Let’s solve for y in the equation x + y = 90.
x + y = 90
Subtract x from both sides.
y = 90 – x
Next, we can substitute 90 – x into the equation y + 2_x_ = 120.
(90 – x) + 2_x_ = 120
90 + x = 120
x = 120 – 90 = 30
x = 30
Since y = 90 – x, y = 90 – 30 = 60.
The question ultimately asks us to find the positive difference between the measures of ACB and CBD. The measure of ACB = 2_x_ – 20 = 2(30) – 20 = 40 degrees. The measure of CBD = y – 10 = 60 – 10 = 50 degrees. The positive difference between 50 degrees and 40 degrees is 10.
The answer is 10.
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Which of the following sets of line-segment lengths can form a triangle?
Which of the following sets of line-segment lengths can form a triangle?
In any given triangle, the sum of any two sides is greater than the third. The incorrect answers have the sum of two sides equal to the third.
In any given triangle, the sum of any two sides is greater than the third. The incorrect answers have the sum of two sides equal to the third.
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In right 
, 
and 
.
What is the value of 
?
In right
What is the value of
There are 180 degrees in every triangle. Since this triangle is a right triangle, one of the angles measures 90 degrees.
Therefore, 
.
There are 180 degrees in every triangle. Since this triangle is a right triangle, one of the angles measures 90 degrees.
Therefore,
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If angle 
and angle 
, what is the value for angle 
?
If angle
For this problem, remember that the sum of the degrees in a triangle is 
.
That means that 
.
Plug in our given values to solve:
Subtract 
from both sides:
For this problem, remember that the sum of the degrees in a triangle is
That means that
Plug in our given values to solve:
Subtract
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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?
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?
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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