Right Triangles - ACT Math

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.

Similar triangles are proportional.

Base1 / Height1 = Base2 / Height2

6 / 9 = 20 / Height2

Cross multiply and solve for Height2

6 / 9 = 20 / Height2

6 * Height2= 20 * 9

Height2= 30

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

To find the area of a triangle use the formula:

, since the base is and the height is , plugging in yields:

Given that it is a right triangle, either angle A or B has to be 90 degrees. The other angle then must be less than 30 degrees, given that C is greater than 60 because there are 180 degrees in a triangle.

Example:

If angle C is 61 degrees and angle A is 90 degrees, then angle B must be 29 degrees in order for the angle measures to sum to 180 degrees.

Since all the answer choices are in trigonometric form, we know we must not necessarily solve for the exact value (although we can do that and calculate each choice to see if it matches). The first step is to determine the length of the ground between the bottom of the ladder and the wall via the Pythagorean Theorem: "x2 + 152 = 172"; x = 8. Using trigonometric definitions, we know that "opposite/adjecent = tan(theta)"; since we have both values of the sides (opp = 15 and adj = 8), we can plug into the tangential form tan(theta) = 15/8. However, since we are solving for theta, we must take the inverse tangent of the left side, "tan-1". Thus, our final answer is

By rule, this is a 3-4-5 right triangle. Sine = (the opposite leg)/(the hypotenuse). This gives us 3/5.

The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.

The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle must be . We know that a right triangle has one angle equal to , and we are told one of the acute angles is .

The rest is simple subtraction:

Thus, our missing angle is .

Solving this problem quickly requires that we recognize how to break apart our ratio.

The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle is . Additionally, the Right Triangle Acute Angle Theorem states that the two non-right angles in a right triangle are acute; that is to say, the right angle is always the largest angle in a right triangle.

Since this is true, we can assume that is represented by the largest number in the ratio of angles. Now consider that the other two angles must also sum to . We know therefore that the sum of their ratios must be divisible by as well.

Thus, .

To find the value of one angle of the ratio, simply assign fractional value to the sum of the ratios and multiply by .

, so:

Thus, the shortest angle (the one represented by in our ratio of angles) is .

The word similar means, comparable in measurement, but not equal. The best sides way to compare these two triangles is by looking at the diagonal side of the triangle since it cannot be mistaken for any of the other sides of either triangle. If the larger triangle has a measurement of 7 and the smaller triangle has a measurement of 3.5 for their diagonal sides, then that means the ratio of the larger triangle to the smaller one is .

This means that segment EF must be similar to segment AC (look at the orientation). So, since segment DE is similar to segment CB, divide 5 by two to get your answer.

1. Find the length of the missing hypotenuse:

You can do this one of two ways:

1. Using the special 3-4-5 right triangle, you can infer that the missing hypotenuse is 5.

2. By using the Pythagorean Theorem, you can solve for the length of the hypotenuse:

In this case:

2. Using the meaning of congruent (the exact same three angles and sides), determine if these two triangles fit this meaning:

They do, because both are 3-4-5 right triangles, and thus must have corresponding and equal angles due to trigonometric properties.

1. Since the two triangles are similar, find the ratio of the two triangles to each other:

In this case, both triangles are multiples of a special right triangle.

and

The ratio of the triangles is

2. Use the ratio you found to solve for , or the length of the missing side:

The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =

32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).

NOTE: For a volume problem: Vnew = (SF)3 x (Vold).

Start by drawing out the light poles and their shadows.

In this case, we end up with two similar triangles. We know that these are similar triangles because the question tells us that these poles are on a flat surface, meaning angle B and angle E are both right angles. Then, because the question states that the shadow cast by both poles are at the same time of day, we know that angles C and F are equivalent. As a result, angles A and D must also be equivalent.

Since these are similar triangles, we can set up proportions for the corresponding sides.

Now, solve for by cross-multiplying.

We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.

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

The base is equal to 6.

The height of an quilateral triangle is equal to , where is the length of the base.

The equation used to find the area of a right triangle is: where A is the area, b is the base, and h is the height of the triangle. In this question, we are given the height, so we need to figure out the base in order to find the area. Since we know both the height and hypotenuse of the triangle, the quickest way to finding the base is using the pythagorean theorem, . a = the height, b = the base, and c = the hypotenuse.

Using the given information, we can write . Solving for b, we get or . Now that we have both the base and height, we can solve the original equation for the area of the triangle.

The easiest way to find the smallest possible integer sides is to simply factor the ratio we are given. In this case, is already prime (since is a prime number), so the smallest possible sides which hold to this triangle are and . You may also recognize this number as a common Pythagorean triple.

The area of a triangle is expressed as , where is the length and is the height. Since our triangle is right, we know that two lines intersect at a angle and thus serve well as our length and height. We also know that the longest side is always the hypotenuse, so the other two sides must be and .

Applying our formula, we get:

Thus, the smallest possible area for our triangle is .