### All Trigonometry Resources

## Example Questions

### Example Question #40 : Right Triangles

Which of the following is true about the right triangle below?

**Possible Answers:**

**Correct answer:**

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.

### Example Question #1 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

**Possible Answers:**

**Correct answer:**

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is . Therefore, .

### Example Question #2 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

**Possible Answers:**

**Correct answer:**

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is . Therefore, .

### Example Question #3 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

**Possible Answers:**

**Correct answer:**

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.

### Example Question #4 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

**Possible Answers:**

**Correct answer:**

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is . Therefore, .

### Example Question #5 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

**Possible Answers:**

The triangle is equilateral.

The triangle is isosceles.

The triangle is scalene.

The triangle is obtuse.

**Correct answer:**

The triangle is isosceles.

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.

### Example Question #6 : Use Special Triangles To Make Deductions

In the figure below, is inscribed in a circle. passes through the center of the circle. In , the measure of is twice the measure of . The figure is drawn to scale.

Which of the following is true about the figure?

**Possible Answers:**

is equal in length to a diameter of the circle.

is equal in length to a radius of the circle.

is equal in length to a diameter of the circle.

is equal in length to a radius of the circle.

**Correct answer:**

is equal in length to a radius of the circle.

For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because is a diameter of the circle, arc has a measure of 180 degrees. Therefore, must be equal to . Since is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of is twice the measure of , . Therefore, the measure of can be calculated as follows:

Therefore, is equal to . must be a 30-60-90 triangle. Therefore, side length must be half the length of side length , the hypotenuse of the triangle. Since is a diameter of the circle, half of represents the length of a radius of the circle. Therefore, is equal in length to a radius of the circle.

### Example Question #7 : Use Special Triangles To Make Deductions

In the figure below, is inscribed in a circle. passes through the center of the circle. In , the measure of is twice the measure of . The figure is drawn to scale.

Which of the following is true about the figure?

**Possible Answers:**

is isosceles.

is a 30-60-90 triangle.

is equilateral.

is a 45-45-90 triangle.

**Correct answer:**

is a 30-60-90 triangle.

For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because is a diameter of the circle, arc has a measure of 180 degrees. Therefore, must be equal to . Since is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of is twice the measure of , . Therefore, the measure of can be calculated as follows:

Therefore, is equal to . must be a 30-60-90 triangle.

### Example Question #8 : Use Special Triangles To Make Deductions

In the figure below, is a diagonal of quadrilateral . has a length of 1. and are congruent and isosceles. and are perpendicular. The figure is drawn to scale.

Which of the following is a true statement?

**Possible Answers:**

is a 30-60-90 triangle.

and , are parallel.

is equilateral.

and are perpendicular.

**Correct answer:**

and , are parallel.

Since and are perpendicular, is a right angle. Since no triangle can have more than one right angle, and is isosceles, must be congruent to . Since is congruent to and measures 90 degrees, and can be calculated as follows:

Therefore, and are both equal to 45 degrees. is a 45-45-90 triangle. Since is congruent to , is also a 45-45-90 triangle. The figure is drawn to scale, so is a right angle. Since has the same angle measure as , the two angles are alternate interior angles and diagonal is a transversal relative to and , which must be parallel.

### Example Question #1 : Use Special Triangles To Make Deductions

In the figure below, is a diagonal of quadrilateral . has a length of . is congruent to .

Which of the following is a true statement?

**Possible Answers:**

The perimeter of quadrilateral is .

The area of quadrilateral is .

The area of quadrilateral is .

The perimeter of quadrilateral is .

**Correct answer:**

The area of quadrilateral is .

Since and are perpendicular, is a right angle. Since no triangle can have more than one right angle, and is isosceles, must be congruent to . Since angle CBD is congruent to and measures 90 degrees, and can be calculated as follows:

Therefore, and are both equal to 45 degrees. is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse is . Anyone of the four side lengths of quadrilateral must, therefore, be equal to . To find the area of , multiply two side lengths: .

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