Equilateral Triangles - GRE Quantitative Reasoning

By having a height in an equilateral triangle, the angle is bisected therefore creating two triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite is and the side of equilateral triangle which is opposite is .

Cross multiply.

Divide both sides by

Multiply top and bottom by to get rid of the radical.

All sides of an equilateral triangle are equal, so all sides of this triangle equal 4.

Area = 1/2 base * height, so we need to calculate the height: this is easy for an equilateral triangle, since you can bisect any such triangle into two identical 30:60:90 triangles.

The ratio of lengths of a 30:60:90 triangle is 1:√3:2. The side of the equilateral triangle is 4, and we divided the base in half when we bisected the triangle, so that give us a length of 2, so our triangle must have sides of 2, 4, and 2√3; thus we have our height.

One of our 30:60:90 triangles will have a base of 2 and a height of 2√3. Half the base is 1, so 1 * 2√3 = 2√3.

We have two of these triangles, since we divided the original triangle, so the total area is 2 * 2√3 = 4√3.

You can also solve for the area of any equilateral triangle by applying the formula (s2√3)/4, where s = the length of any side.

An equilateral triangle can be considered to be 2 identical 30-60-90 triangles, giving the triangle a height of 6$\sqrt{3}$. From there, use the formula for the area of a triangle:

To solve this equation, first note that a line drawn from the origin to a vertex of the equilateral triangle will bisect the angle of the vertex. Furthermore, the length of this line is equal to the radius:

That this creates in turn is a 30-60-90 right triangle. Recall that the ratio of the sides of a 30-60-90 triangle is given as:

Therefore, the length of the side can be found to be

This is also one half of the base of the triangle, so the base of the triangle can be found to be:

Furthermore, the length of the side is:

The vertical section rising from the origin is the length of the radius, which when combined with the shorter section above gives the height of the triangle:

The area of a triangle is given by one half the base times the height, so we can find the answer as follows:

Perimeter is found by adding up all sides of the triangle. All sides in an equilateral triangle are equal, so we need to find the value of just one side to know the values of all sides.

The height of an equilateral triangle divides it into two equal 30:60:90 triangles, which will have side ratios of 1:2:√3. The height here is the √3 ratio, which in this case is equivalent to 8, so to get the length of the other two sides, we put 8 over √3 (8/√3) and 2 * 8/√3 = 16/√3, which is the hypotenuse of our 30:60:90 triangle.

The perimeter is then 3 * 16/√3, or 48/√3.

By having a height in an equilateral triangle, the angle is bisected therefore creating two triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite is and the side of equilateral triangle which is opposite is .

Cross multiply.

Divide both sides by

Multiply top and bottom by to get rid of the radical.

Since each side is the same and there are three sides, we just multply the answer by three to get .

The area of an equilateral triangle is .

So let's set-up an equation to solve for .

Cross multiply.

The cancels out and we get .

Then take square root on both sides and we get . Since we have three equal sides, we just multply by three to get as the final answer.

Perimeter is found by adding up all sides of the triangle. All sides in an equilateral triangle are equal, so we need to find the value of just one side to know the values of all sides.

The height of an equilateral triangle divides it into two equal 30:60:90 triangles, which will have side ratios of 1:2:√3. The height here is the √3 ratio, which in this case is equivalent to 8, so to get the length of the other two sides, we put 8 over √3 (8/√3) and 2 * 8/√3 = 16/√3, which is the hypotenuse of our 30:60:90 triangle.

The perimeter is then 3 * 16/√3, or 48/√3.

By having a height in an equilateral triangle, the angle is bisected therefore creating two triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite is and the side of equilateral triangle which is opposite is .

Cross multiply.

Divide both sides by

Multiply top and bottom by to get rid of the radical.

Since each side is the same and there are three sides, we just multply the answer by three to get .

The area of an equilateral triangle is .

So let's set-up an equation to solve for .

Cross multiply.

The cancels out and we get .

Then take square root on both sides and we get . Since we have three equal sides, we just multply by three to get as the final answer.

All sides of an equilateral triangle are equal, so all sides of this triangle equal 4.

Area = 1/2 base * height, so we need to calculate the height: this is easy for an equilateral triangle, since you can bisect any such triangle into two identical 30:60:90 triangles.

The ratio of lengths of a 30:60:90 triangle is 1:√3:2. The side of the equilateral triangle is 4, and we divided the base in half when we bisected the triangle, so that give us a length of 2, so our triangle must have sides of 2, 4, and 2√3; thus we have our height.

One of our 30:60:90 triangles will have a base of 2 and a height of 2√3. Half the base is 1, so 1 * 2√3 = 2√3.

We have two of these triangles, since we divided the original triangle, so the total area is 2 * 2√3 = 4√3.

You can also solve for the area of any equilateral triangle by applying the formula (s2√3)/4, where s = the length of any side.

An equilateral triangle can be considered to be 2 identical 30-60-90 triangles, giving the triangle a height of 6$\sqrt{3}$. From there, use the formula for the area of a triangle:

To solve this equation, first note that a line drawn from the origin to a vertex of the equilateral triangle will bisect the angle of the vertex. Furthermore, the length of this line is equal to the radius:

That this creates in turn is a 30-60-90 right triangle. Recall that the ratio of the sides of a 30-60-90 triangle is given as:

Therefore, the length of the side can be found to be

This is also one half of the base of the triangle, so the base of the triangle can be found to be:

Furthermore, the length of the side is:

The vertical section rising from the origin is the length of the radius, which when combined with the shorter section above gives the height of the triangle:

The area of a triangle is given by one half the base times the height, so we can find the answer as follows:

Perimeter is found by adding up all sides of the triangle. All sides in an equilateral triangle are equal, so we need to find the value of just one side to know the values of all sides.

The height of an equilateral triangle divides it into two equal 30:60:90 triangles, which will have side ratios of 1:2:√3. The height here is the √3 ratio, which in this case is equivalent to 8, so to get the length of the other two sides, we put 8 over √3 (8/√3) and 2 * 8/√3 = 16/√3, which is the hypotenuse of our 30:60:90 triangle.

The perimeter is then 3 * 16/√3, or 48/√3.

By having a height in an equilateral triangle, the angle is bisected therefore creating two triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite is and the side of equilateral triangle which is opposite is .

Cross multiply.

Divide both sides by

Multiply top and bottom by to get rid of the radical.

Since each side is the same and there are three sides, we just multply the answer by three to get .

The area of an equilateral triangle is .

So let's set-up an equation to solve for .

Cross multiply.

The cancels out and we get .

Then take square root on both sides and we get . Since we have three equal sides, we just multply by three to get as the final answer.

The area of an equilateral triangle is .

So let's set-up an equation to solve for .

Cross multiply.

The cancels out and we get .

Then take square root on both sides and we get . To find height, we need to realize by drawing a height we create triangles.

The height is opposite the angle . We can set-up a proportion. Side opposite is and the side of equilateral triangle which is opposite is .

Cross multiply.

Divide both sides by

We can simplify this by factoring out a to get a final answer of .

To find the area of an equilateral triangle, notice that it can be divided into two triangles:

The ratio of sides in a triangle is , and since the triangle is bisected such that the degree side is , the degree side, the height of the triangle, must have a length of .

The formula for the area of the triangle is given as:

So the area of an equilateral triangle can be written in term of the lengths of its sides as:

For this particular triangle, since , its area is equal to .

If the relation between ratios is hard to visualize, realize that

This problem requires a bit of creative thinking (unless you have memorized the fact that an equilateral triangle always has an area equal to its side length times .

Consider the equilateral triangle:

Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:

This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:

We can also say, given our figure, that the following equivalence must hold:

Solving for , we get:

Now, since , we know that must be smaller than . This means that or . Quantity B is larger than quantity A.