No two of the sides are of equal length, making the triangle scalene.

To determine whether it is acute or obtuse, add the squares of the lesser lengths, and compare the sum to the square of the greatest length:

The sum of the squares of the lesser lengths exceeds the square of the greatest length, so the triangle is acute.

The correct response is that the triangle is scalene and acute. 

No two of the sides are of equal length, making the triangle scalene.

To determine whether it is acute or obtuse, add the squares of the lesser lengths, and compare the sum to the square of the greatest length:

The square of the greatest length exceeds the sum of the squares of the lesser lengths, so the triangle is obtuse.

The correct response is that the triangle is scalene and obtuse.

The perimeter of any figure is the sum of all the lengths of that figure.

An equilateral triangle is a triangle where the length of all three sides is the same length. So, all sides of the triangle have a length of cm. 

So, the perimeter is , or cm.

First we need to know that in an isosceles triangle, 2 sides are equal in size so we know that since one side is 8cm, the other side is also 8cm.

We are also given information that the base is half the length of the size, and half of 8 is 4, so our base is 4cm.

The perimeter of any triangle is found when adding up all sides.

In this case we add up 

The answer is independent of the sidelengths of the rectangle, so to ease calculations, we will arbitrarily assign to Square  sidelength 4 inches. 

The shaded region can be divided by a perpendicular segment from  to , then another from  to that segment, as follows:

By the fact that rectangles have opposite sides that are congruent, and by the fact that  is the midpoint of and  is the midpoint of , the shaded region is a composite of:

Square , which has sides of length 2 and therefore area ;

Right triangle , which has two legs of length 2 and, thus, area ; and,

Right triangle , which has legs of length 4 and 2 and, thus, area .

The area of the shaded region is therefore .

Square  has area , so the shaded region is

of the square.

This triangle is isosceles by the converse of the Isosceles Triangle Theorem. The altitude of the triangle divides it into two triangles that are both right and that are congruent, as follows:

Each triangle is a  triangle. The altitude is the short leg of each triangle, and, therefore, its length is half that of the common hypotenuse, or 6. The long leg of each right triangle has length  times that of the short leg, or , and the base of the entire large triangle is twice this, or . 

The area of the triangle is half the product of the height and the base:

The area of a triangle is half the product of its baselength and its height. 

To find the area of , we can use the lengths of the legs  and :

To find the area of , we can use the hypotenuse , the length of which is 30, and the altitude  perpendicular to it:

In terms of area,  is

of .

The inscribed rectangle is a 20 by 20 square. Since opposite sides of the square are parallel, the corresponding angles of the two smaller right triangles are congruent; therefore, the two triangles are similar and, by definition, their sides are in proportion.

The small top triangle has legs 10 and 20; the blue triangle has legs 20 and , where  can be calculated with the following proportion:

The legs of the blue triangle are 20 and 40; half their product is the area:

To find the area of a triangle, we will use the following formula:

where b  is the base and h is the height of the triangle.

Now, we know the base of the triangle is 7in. We also know the height of the triangle is two times the base. Therefore, the height is 14in.  So, we can substitute. We get

To find the area of a triangle, we will use the following formula:

where b is the base and h is the height of the triangle. 

Now, we know the base of the triangle is 8in. We also know the height of the triangle is half the base. Therefore, the height is 4in. So, we can substitute. We get