Calculating the area of a right triangle - GMAT Quantitative

This is a right triangle with legs along the - and -axes, so the area of each can be calculated by taking one-half the product of the two legs.

The vertical leg has length ; the horizontal leg has length .

Now calculate the area:

Since are right triangles, we know that is an isosceles right triangle. So we know that the lengths of and are 2 cm, so we can get the length of by using the Pythagorean Theorem:

is the midpoint of , so the length of is .

Now we can use the Pythagorean Theorem again to solve for : .

Finally, we have all the elements needed to solve for the area of :

Let be the sidelength of the square. Then its area is .

If the hypotenuse of a triangle is , its shorter leg is half that, or ; its longer leg is times the shorter leg, or . The area of the triangle is half the product of the legs, or

The ratio of the area of the square to that of the triangle is

or

or

The equation for the area of a right triangle is:

In this case, our values are:

Plugging this into the equation leaves us with:

which can be rewritten as

In order to calculate the area of a right triangle, we need to know the height and the base. We are only given the length of the base, so we first need to use the Pythagorean theorem with the given hypotenuse to find the length of the height:

Now that we have the height and the base, we can plug these values into the formula for the area of a right triangle:

To find the area, use the following formula:

Since are right triangles, we know that is an isosceles right triangle. So we know that the lengths of and are 2 cm, so we can get the length of by using the Pythagorean Theorem:

is the midpoint of , so the length of is .

Now we can use the Pythagorean Theorem again to solve for : .

Finally, we have all the elements needed to solve for the area of :

Becaue this is a right triangle and since the hypotenuse is 10 and one side is 6, the other side will be 8. This can be found using the Pythagorean Theorem, or multiplying a 3-4-5 triangle by 2 to get a 6-8-10 triangle. Since the area of a right triangle is half of the product of the two sides, we have

This is a right triangle with legs along the - and -axes, so the area of each can be calculated by taking one-half the product of the two legs.

The vertical leg has length ; the horizontal leg has length .

Now calculate the area:

Let be the sidelength of the square. Then its area is .

If the hypotenuse of a triangle is , its shorter leg is half that, or ; its longer leg is times the shorter leg, or . The area of the triangle is half the product of the legs, or

The ratio of the area of the square to that of the triangle is

or

or

The equation for the area of a right triangle is:

In this case, our values are:

Plugging this into the equation leaves us with:

which can be rewritten as

In order to calculate the area of a right triangle, we need to know the height and the base. We are only given the length of the base, so we first need to use the Pythagorean theorem with the given hypotenuse to find the length of the height:

Now that we have the height and the base, we can plug these values into the formula for the area of a right triangle:

To find the area, use the following formula:

Becaue this is a right triangle and since the hypotenuse is 10 and one side is 6, the other side will be 8. This can be found using the Pythagorean Theorem, or multiplying a 3-4-5 triangle by 2 to get a 6-8-10 triangle. Since the area of a right triangle is half of the product of the two sides, we have

This is a right triangle with legs along the - and -axes, so the area of each can be calculated by taking one-half the product of the two legs.

The vertical leg has length ; the horizontal leg has length .

Now calculate the area:

Since are right triangles, we know that is an isosceles right triangle. So we know that the lengths of and are 2 cm, so we can get the length of by using the Pythagorean Theorem:

is the midpoint of , so the length of is .

Now we can use the Pythagorean Theorem again to solve for : .

Finally, we have all the elements needed to solve for the area of :

Let be the sidelength of the square. Then its area is .

If the hypotenuse of a triangle is , its shorter leg is half that, or ; its longer leg is times the shorter leg, or . The area of the triangle is half the product of the legs, or

The ratio of the area of the square to that of the triangle is

or

or

The equation for the area of a right triangle is:

In this case, our values are:

Plugging this into the equation leaves us with:

which can be rewritten as

In order to calculate the area of a right triangle, we need to know the height and the base. We are only given the length of the base, so we first need to use the Pythagorean theorem with the given hypotenuse to find the length of the height:

Now that we have the height and the base, we can plug these values into the formula for the area of a right triangle: