Geometry Chapter 9 Test

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem. To determine if the side lengths form a Pythagorean triple, check if the square of the longest side is equal to the sum of the squares of the other two sides.

The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

The angle of elevation is the angle between the horizontal and the line of sight to an object above the horizontal.

To solve a right triangle means to find the lengths of all the sides and the measures of all the angles. This can be done by using trigonometric ratios, the Pythagorean Theorem, and the fact that the sum of the angles in a triangle is 180 degrees.

The measure of angle B is 90 degrees because it forms a right angle.

The length of side b is 12 units. This can be found using the Pythagorean Theorem because it's the hypotenuse of a right triangle.

The length of side c is 16 units. You can find this using the Pythagorean Theorem or by recognizing the 3-4-5 right triangle pattern.

To make triangles ABC and DEF similar, the ratio of corresponding sides must be equal. In this case, x must be 9 to make the ratio of corresponding sides equal.

The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Therefore, ∠1 ≅ ∠2.

The measure of angle JKM is 40 degrees. This can be determined using the fact that adjacent angles are supplementary - they add up to 180 degrees.

AD is 3 units long because the smaller triangle ADE is similar to the larger triangle ABC. This allows us to find the proportional sides.

The length of DE is 4 units. This can be determined using the information from the similar triangles ADE and ABC, and the relationship between corresponding sides.

The ratio of the opposite side to the adjacent side in a right triangle.

The ratio of the opposite side to the hypotenuse in a right triangle.

A right triangle is a triangle with one angle that measures 90 degrees. Given the measurements of two sides or one side and one angle (excluding the right angle), you can determine the remaining sides and angles using trigonometric ratios.

The height of the tree is the opposite side, the distance from the observer to the tree is the adjacent side, and the angle of elevation is the known angle. Use the tangent ratio to find the height.

The distance x is the adjacent side to the known angle, and the distance to the top of the mountain is the hypotenuse. Use the cosine function to determine the distance x.

Create a diagram using the given information and form two triangles. Use Law of Cosines to find the distance between Cities B & C. Remember to use the correct angle and sides.

The Law of Cosines is a generalization of the Pythagorean theorem that applies to any triangle. It allows you to find side lengths and angles, even if the triangle is not a right triangle.

The trigonometric functions (sine, cosine, tangent, etc.) are used to relate the angles and sides of a right triangle. These ratios are important for solving problems involving triangles, especially those related to navigation, surveying, and engineering.