This is a  triangle. We can find the value of the other leg by using the Pythagorean Theorem.

Remembering that

.

Thus,

.

If , you know the adjacent side is .

Thus, making

 because tangent is opposite/adjacent.

Given two angles of unequal measure in a triangle, the side opposite the greater angle is longer than the side opposite the other angle. Since , it follows that . Also, in a right triangle, the hypotenuse must be the longest side; in , since  is the right side, this hypotenuse is . It follows that , and that (I) is the only statement that can possibly be true.

AB is the leg adjacent to Angle A and BC is the leg opposite Angle A.

Since we have a  triangle, the opposites sides of those angles will be in the ratio .

Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to .

which also means

The ladder leaning against the wall forms a right triangle. The hypotenuse of the triangle is 5 ft., the length of the ladder. 

Because sin x= opposite/hypotenuse, sine of the angle is equal to the length of the side opposite the angle divided by the hypotenuse. In this case, the length of the side opposite the angle is h, the height of the end of the ladder that is touching the wall. Thus,

Because we are told that 

we know that h=3. Therefore, 3 feet is the height of the ladder. If the ladder is falling at a rate of 6 feet per second, we can find the number of seconds it will take the ladder to hit the ground with the equation

where h represents the height the ladder is falling from, and s represents the number of seconds it takes the ladder to fall. We can now solve for s: 

It takes the ladder 0.5 seconds to fall to the ground. 

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 20. Therefore,

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

Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.

Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.

We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.

We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:

 a² + b² = c²

15² + 20² = c²

625 = c²

c = 25

The length of BD is 25.

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of AB, BC, CD, and DA.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85. 

We can set this problem up like a set of similar triangles.  

The first triangle is created by the three points: The Traffic light, the spot beneath the traffic light, and the spot where the man is standing (which is also the spot where the traffic light's shadow is).

The height of this Triangle is "T" as given in the question, and its base is the part that we are asked to solve for.

The second triangle is created by the top of the man's head, his feet, and the end of his shadow.

The height of this Triangle is "h" as given in the questions, and the base is "s".

We set up a proportion:

where X is the distance we are asked to find.  Simply cross-multiply to solve.

To calculate the width of the crater, use the given information to establish that the image draws similar triangles. When triangles that have corresponding angles and a ratio to their side lengths they are considered to be similar triangles.

Identify the known information.

therefore,

and the bases of the triangles are parallel.

Also,

Set up the side ratios for this particular problem.

Looking at the only full ratio that is given, the scalar multiplier can be found.

Therefore, to find the width of the crater  multiply  by two.

By the Pythagorean Theorem, 

Set , and solve for : 

Take the positive square root of both sides:

, so corresponding sides are in proportion; specifically, 

Set , and solve for :

A right triangle has as its area half the product of the length of its legs, so the area of  is

To the nearest whole number, this rounds to 35.

There are 2 components to this problem. The first, and easier one, is recognizing how much of the perimeter the equilateral triangles take up—since there are 40 triangles in total, there must be 40 – 4 = 36 of these triangles. By observation, each contributes only 1 side to the overall perimeter, thus we can simply multiply 36(5) = 180" contribution.

The second component is the corner triangles—recognizing that the congruent sides are adjacent to the 5-inch equilateral triangles, and the congruent angles can be found by

180 = 30+2x → x = 75°

We can use ratios to find the unknown side:

75/5 = 30/y → 75y = 150 → y = 2''.

Since there are 4 corners to the square rug, 2(4) = 8'' contribution to the total perimeter. Adding the 2 components, we get 180+8 = 188 inch perimeter.