Honors Geometry Fall Final Review Quiz 2025

1) Write the equation of a line that is perpendicular to y = 3x - 6 and passes through the point (-5, 2).

Y = -1/3x - 3/3

Y = 3x + 3

Y = -1/3x + 3

Y = 3x - 6

To find the equation of a line perpendicular to \( y = 3x - 6 \), we first determine its slope, which is 3. The slope of a line perpendicular to it is the negative reciprocal, yielding \( -\frac{1}{3} \). Using the point-slope form of a line, we substitute the point (-5, 2) into the equation \( y - y_1 = m(x - x_1) \). This gives us \( y - 2 = -\frac{1}{3}(x + 5) \). Simplifying this leads to the equation \( y = -\frac{1}{3}x + 3 \).

Explanation

To find the equation of a line perpendicular to \( y = 3x - 6 \), we first determine its slope, which is 3. The slope of a line perpendicular to it is the negative reciprocal, yielding \( -\frac{1}{3} \). Using the point-slope form of a line, we substitute the point (-5, 2) into the equation \( y - y_1 = m(x - x_1) \). This gives us \( y - 2 = -\frac{1}{3}(x + 5) \). Simplifying this leads to the equation \( y = -\frac{1}{3}x + 3 \).

2) In the coordinate plane, if line n has a slope of 2, what is the slope of a line perpendicular to line n?

-1/2

1/2

2

-2

In the coordinate plane, the slopes of two perpendicular lines are negative reciprocals of each other. If line n has a slope of 2, the negative reciprocal is calculated by taking the negative of the reciprocal of 2. This is done by flipping the fraction (1/2) and changing its sign, resulting in -1/2. Therefore, the slope of a line that is perpendicular to line n is -1/2.

Explanation

In the coordinate plane, the slopes of two perpendicular lines are negative reciprocals of each other. If line n has a slope of 2, the negative reciprocal is calculated by taking the negative of the reciprocal of 2. This is done by flipping the fraction (1/2) and changing its sign, resulting in -1/2. Therefore, the slope of a line that is perpendicular to line n is -1/2.

3) What is the sum of the interior angles of a triangle?

180 degrees

360 degrees

90 degrees

270 degrees

The sum of the interior angles of a triangle is always 180 degrees, regardless of the type of triangle. This is a fundamental property in Euclidean geometry, derived from the fact that a triangle can be divided into two right triangles by drawing a line from one vertex to the opposite side. Each of these right triangles contributes to the total angle sum, leading to the conclusion that all triangles, whether scalene, isosceles, or equilateral, will always have their interior angles add up to 180 degrees.

Explanation

The sum of the interior angles of a triangle is always 180 degrees, regardless of the type of triangle. This is a fundamental property in Euclidean geometry, derived from the fact that a triangle can be divided into two right triangles by drawing a line from one vertex to the opposite side. Each of these right triangles contributes to the total angle sum, leading to the conclusion that all triangles, whether scalene, isosceles, or equilateral, will always have their interior angles add up to 180 degrees.

4) If a parallelogram has one angle measuring 70 degrees, what is the measure of the opposite angle?

70 degrees

110 degrees

90 degrees

180 degrees

In a parallelogram, opposite angles are congruent, meaning they have the same measure. Since one angle is given as 70 degrees, the angle directly opposite to it must also be 70 degrees. This property ensures that both pairs of opposite angles in a parallelogram are equal, confirming that if one angle measures 70 degrees, the opposite angle will also measure 70 degrees.

Explanation

In a parallelogram, opposite angles are congruent, meaning they have the same measure. Since one angle is given as 70 degrees, the angle directly opposite to it must also be 70 degrees. This property ensures that both pairs of opposite angles in a parallelogram are equal, confirming that if one angle measures 70 degrees, the opposite angle will also measure 70 degrees.

5) Which of the following statements is true about the diagonals of a parallelogram?

They are always equal.

They bisect each other.

They are perpendicular.

They are congruent.

In a parallelogram, the diagonals intersect at their midpoints, meaning they divide each other into two equal segments. This property is a fundamental characteristic of parallelograms, distinguishing them from other quadrilaterals. While the diagonals may not be equal in length or perpendicular, the fact that they bisect each other holds true for all parallelograms, confirming this statement as correct.

Explanation

In a parallelogram, the diagonals intersect at their midpoints, meaning they divide each other into two equal segments. This property is a fundamental characteristic of parallelograms, distinguishing them from other quadrilaterals. While the diagonals may not be equal in length or perpendicular, the fact that they bisect each other holds true for all parallelograms, confirming this statement as correct.

6) What is the formula for the area of a triangle?

1/2 * base * height

Base * height

Base + height

1/3 * base * height

The area of a triangle is calculated using the formula \( \frac{1}{2} \times \text{base} \times \text{height} \). This formula derives from the fact that a triangle can be viewed as half of a rectangle formed by its base and height. By multiplying the base by the height, you find the area of the rectangle, and since a triangle occupies half that space, the area is halved, leading to the formula. This relationship holds true for all triangles, regardless of their orientation or type.

Explanation

The area of a triangle is calculated using the formula \( \frac{1}{2} \times \text{base} \times \text{height} \). This formula derives from the fact that a triangle can be viewed as half of a rectangle formed by its base and height. By multiplying the base by the height, you find the area of the rectangle, and since a triangle occupies half that space, the area is halved, leading to the formula. This relationship holds true for all triangles, regardless of their orientation or type.

7) If the coordinates of a point are (3, 4), what is the distance from the origin (0, 0) to this point?

5

7

4

3

To find the distance from the origin (0, 0) to the point (3, 4), we use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the coordinates, we have \( d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). Thus, the distance is 5 units.

Explanation

To find the distance from the origin (0, 0) to the point (3, 4), we use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the coordinates, we have \( d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). Thus, the distance is 5 units.

8) In a right triangle, if one angle measures 30 degrees, what is the measure of the other non-right angle?

60 degrees

30 degrees

90 degrees

45 degrees

In a right triangle, the sum of all angles is always 180 degrees. Since one angle is a right angle (90 degrees) and another angle is given as 30 degrees, we can find the measure of the third angle by subtracting the sum of these two angles from 180 degrees. Therefore, 180 - 90 - 30 = 60 degrees. Thus, the measure of the other non-right angle is 60 degrees.

Explanation

In a right triangle, the sum of all angles is always 180 degrees. Since one angle is a right angle (90 degrees) and another angle is given as 30 degrees, we can find the measure of the third angle by subtracting the sum of these two angles from 180 degrees. Therefore, 180 - 90 - 30 = 60 degrees. Thus, the measure of the other non-right angle is 60 degrees.

9) What is the Pythagorean theorem?

A^2 + b^2 = c^2

A + b = c

A^2 - b^2 = c^2

A + b^2 = c

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This fundamental relationship is essential in geometry, allowing for the calculation of one side of a right triangle when the lengths of the other two sides are known. It is widely used in various fields, including architecture, engineering, and physics, to solve problems involving right-angled triangles.

Explanation

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This fundamental relationship is essential in geometry, allowing for the calculation of one side of a right triangle when the lengths of the other two sides are known. It is widely used in various fields, including architecture, engineering, and physics, to solve problems involving right-angled triangles.

10) If a triangle has sides of lengths 3, 4, and 5, what type of triangle is it?

Right triangle

Acute triangle

Obtuse triangle

Equilateral triangle

A triangle with sides of lengths 3, 4, and 5 can be classified using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Here, \(5^2 = 25\) and \(3^2 + 4^2 = 9 + 16 = 25\). Since this condition holds true, the triangle is confirmed to be a right triangle.

Explanation

A triangle with sides of lengths 3, 4, and 5 can be classified using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Here, \(5^2 = 25\) and \(3^2 + 4^2 = 9 + 16 = 25\). Since this condition holds true, the triangle is confirmed to be a right triangle.