The formula for the area of a triangle is , where is the length and is the height of the triangle. In this case, we know that , which means . Since we are told that the sides are integers, this means that must consist of two factors of . Since is the only number choice not a factor of , it is the only answer that could not meet the requirements.

To solve, simply realize this is one of our special right triangles, . Thus, our answer is . If you forgot this, you can use the Pythagorean Theorem.

1. Use the pythagorean theorem with and :

2. Solve for :

(Notice that this right triangle is also a triangle.)

To answer this, we first need to understand that we are trying to find the leg of a right triangle. The ladder, when leaned against the wall, forms a right triangle where the ladder is the hypotenuse, the wall is one leg, and the ground between the ladder and the wall is the other leg.

To find how high up the ladder touches the wall, we use the Pythagorean Theorem, which is:

is the length of the hypotenuse (the ladder itself), is the distance of the ladder base from the wall, and is how high the ladder touches on the wall.

We then rearrange to solve for one of the legs by subtracting from both sides.

We can now plug in our values of 10 for the hypotenuse and one of our legs (in this case, )

We can then take the square root of this equation to get an answer for

The question asked us to approximate, so we must round to the nearest whole number. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

Therefore, our answer is .