Assigning the length of ED the value of x, the value of AE will be 3_x_. That makes the entire side AD equal to 4_x_. Since the figure is a square, all four sides will be equal to 4_x_. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3_x_, 4_x_ and 10. Using the Pythagorean theorem:

(3_x_)2 + (4_x_)2 = 102

9_x_2 + 16_x_2 = 100

25_x_2 = 100

_x_2 = 4

x = 2

With x = 2, each side of the square is 4_x_, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.

Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100Ÿ√2=c. 10√2=c

If we know two sides are equal to and we know that one of the angles is a right angle, then that means that this must be a Special Right Triangle where the interior angles are .

With this special triangle, we also know that the measure of the hypotenuse is equal to the measure of one side of the triangle times the square root of that measure.

Since one leg of the triangle is . then the hypotenuse is equal to .

We could also solve this using the Pythagorean Theorem, like so:

The Pythagorean Theorem states that . This question gives us the values of and , and asks us to solve for .

Take and and plug them into the equation as and :

Now we can start solving for :

The length of the hypotenuse is .

The triangle has two sides of equal length, 13, so it is by definition isosceles.

To determine whether the triangle is acute, right, or obtuse, compare the sum of the squares of the lengths of the two shortest sides to the square of the length of the longest side. The former quantity is equal to

The latter quantity is equal to

The former is greater than the latter; consequently, the triangle is acute. The correct response is that the triangle is acute and isosceles.

The Pythagorean Theorem states that . This question gives us the values of and , and asks us to solve for .

Take and and plug them into the equation as and :

Now we can start solving for :

The length of the hypotenuse is .

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is , so

Substituting for :

This makes a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg is equal to that of hypotenuse , the length of which is 12, divided by . Therefore,

Rationalize the denominator by multiplying both halves of the fraction by :

By the same reasoning, .

The perimeter of the triangle is

Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes .

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is , so

Substituting for :

This makes a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg is equal to that of hypotenuse , the length of which is 12, divided by . Therefore,

Rationalize the denominator by multiplying both halves of the fraction by :

By the same reasoning, .

The perimeter of the triangle is