Triangles - Math

Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.

The Pythagorean theorem is a2 + b2 = c2, where a and b are legs of the right triangle, and c is the hypotenuse.

(15)2 + (20)2 = c2 so c2 = 625. Take the square root to get c = 25m

Similar triangles are in proportion.

Use Pythagorean Theorem to solve for AC:

Pythagorean Theorem: _AB_2 + _BC_2 = _AC_2

42 + 32 = _AC_2

16 + 9 = _AC_2

25 = _AC_2

AC = 5

If the similar triangle's hypotenuse is 25, then the proportion of the sides is AC/25 or 5/25 or 1/5.

Two legs then are 5 times longer than AB or BC:

5 * (AB) = 5 * (4) = 20

5 * (BC) = 5 * (3) = 15

The area of a triangle is denoted by the equation 1/2 b x h.

b stands for the length of the base, and h stands for the height.

Here we are told that the perimeter (total length of all three sides) is 12, and the hypotenuse (the side that is neither the height nor the base) is 5 units long.

So, 12-5 = 7 for the total perimeter of the base and height.

7 does not divide cleanly by two, but it does break down into 3 and 4,

and 1/2 (3x4) yields 6.

Another way to solve this would be if you recall your rules for right triangles, one of the very basic ones is the 3,4,5 triangle, which is exactly what we have here

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

By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:

102 + 52 = _x_2

100 + 25 = _x_2

√125 = x, but we still need to factor the square root

√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so

5√5= x

Using the Pythagorean theorem, _x_2 + _y_2 = _h_2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = _h_2 . Multiplied out 25 + 25 = _h_2.

Therefore _h_2 = 50, so h = √50 = √2 * √25 or 5√2.

First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(12 + 12) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.

To answer this question without plugging all five answer choices in to the Pythagorean Theorem (which takes too long on the GRE), we can use special triangle formulas. Remember that 45-45-90 triangles have lengths of x, x, x√2. Similarly, 30-60-90 triangles have lengths x, x√3, 2x. We should also recall that 3,4,5 and 5,12,13 are special right triangles. Therefore the set of sides that doesn't fit any of these rules is 6, 7, 8.

This can be solved with the Pythagorean Theorem.

62 + 42 = _c_2

52 = _c_2

c = √52 = 2√13

By using the Pythagorean Theorem, we can solve for the distance “as the crow flies” from Paul to his home:

32 + 42 = _x_2

9 + 16 = _x_2

25 = _x_2

5 = x

Because we are told this is a right triangle, we can use the Pythagorean Theorem, _a_2 + _b_2 = _c_2. You may also remember some of these as special right triangles that are good to memorize, such as 3, 4, 5.

Here, 6, 8, 11 will not be the sides to a right triangle because 62 + 82 = 102.

The hypotenuse can be found using Pythagorean Theorem, which is a2 + b2 = c2, so we plug in a = 3 and b = 4 to get c.

c2 =25, so c = 5

Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.

_a_2 + _b_2 = _c_2

52 + 82 = _c_2

25 + 64 = _c_2

89 = _c_2

c = √89

An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).

The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).

5 + 5 + 6 = 16

To find the area of a triangle, multiply the base by the height, then divide by 2. Since the short legs of an isosceles triangle are the same length, we need to know only one to know the other. Since, a short side serves as the base of the triangle, the other short side tells us the height.

In order to calculate the triangle's area, we need to find the lengths of its legs. An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.

Now we can calculate the area using the formula

Now, convert to feet.

This problem represents the definition of the side lengths of an isosceles right triangle. By definition the sides equal , , and . However, if you did not remember this definition one can also find the length of the side using the Pythagorean theorem .

To find the length of the diagonal, given two sides of the square, we can create two equal triangles from the square. The diagonal line splits the right angles of the square in half, creating two triangles with the angles of , , and degrees. This type of triangle is a special right triangle, with the relationship between the side opposite the degree angles serving as x, and the side opposite the degree angle serving as .

Appyling this, if we plug in for we get that the side opposite the right angle (aka the diagonal) is

If the area of the square is , we know that each side of the square is , because the area of a square is .

Then, the diagonal creates two special right triangles. Knowing that the sides = , we can find that the hypotenuse (aka diagonal) is