How to find the height of an equilateral triangle - GRE Quantitative Reasoning

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.

If the perimeter of our equilateral triangle is , each of its sides must be or . This gives us the following figure:

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:

Therefore, 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

Therefore, quantity B is larger than quantity A.

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.

If the perimeter of our equilateral triangle is , each of its sides must be or . This gives us the following figure:

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:

Therefore, 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

Therefore, quantity B is larger than quantity A.

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.

If the perimeter of our equilateral triangle is , each of its sides must be or . This gives us the following figure:

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:

Therefore, 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

Therefore, quantity B is larger than quantity A.

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.

If the perimeter of our equilateral triangle is , each of its sides must be or . This gives us the following figure:

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:

Therefore, 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

Therefore, quantity B is larger than quantity A.

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.

If the perimeter of our equilateral triangle is , each of its sides must be or . This gives us the following figure:

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:

Therefore, 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

Therefore, quantity B is larger than quantity A.