Geometry for US Curriculum
Geometry is considered to be the language of God. This is really wonderful that our creator is the master of geometry. The Geometry help us to visualize the shapes. It also enhance our critical thinking, it helps us to develop our logic and reasoning. Noble Learners has best Geometry tutoring service for the US curriculum. Usually, Geometry is taught in Grade-9 or Grade-10 in the US depending on the State and School. In case of accelerated Math, one can also learn Geometry in Grade-8 as well. Best Online Tutors for Geometry from India are available on Noble Learners. Book a free demo now and start your learning journey today.
Geometry Syllabus
The chapters given below is the part of syllabus which we will discuss in course. In case of any variation in the syllabus our expert tutors from India will teach as per the need of the course from one student to another.
Chapter 1: Reasoning in Geometry
Chapter 2: Segment Measure and Coordinate Graphing
Chapter 3: Angles
Chapter 4: Parallels
Chapter 5: Triangles and Congruence
Chapter 6: More About Triangles
Chapter 7: Triangle Inequalities
Chapter 8: Quadrilaterals
Chapter 9: Proportions and Similarity
Chapter 10: Polygons and Area
Chapter 11: Circles
Chapter 12: Surface Area and Volume
Chapter 13: Right Triangles and Trigonometry
Chapter 14: Circle Relationships
Chapter 15: Formalizing Proof
Chapter 16: More Coordinate Graphing and Transformations
Our Features
Chapter 1: Reasoning in Geometry
- Question: State the definition of a polygon.
Answer: A polygon is a closed figure formed by three or more line segments. - Question: Identify and define the different types of angles in a polygon.
Answer: Interior angles, exterior angles, and central angles. - Question: Explain the difference between convex and concave polygons.
Answer: A convex polygon has all interior angles less than 180 degrees, while a concave polygon has at least one interior angle greater than 180 degrees. - Question: Define the term "congruence" in geometry.
Answer: Congruence means that two figures have the same shape and size. - Question: State the triangle inequality theorem.
Answer: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Chapter 2: Segment Measure and Coordinate Graphing
- Question: Find the midpoint of the line segment with endpoints (3, 4) and (7, 10).
Answer: Midpoint = \( \left(\frac{3 + 7}{2}, \frac{4 + 10}{2}\right) = (5, 7) \). - Question: Calculate the length of the line segment with endpoints (2, 5) and (8, 9).
Answer: Distance = \( \sqrt{(8 - 2)^2 + (9 - 5)^2} = \sqrt{36 + 16} = \sqrt{52} \). - Question: Plot the points (3, 2), (-4, 5), and (0, -3) on a coordinate plane.
Answer: Points should be plotted as described. - Question: Determine the coordinates of the point that divides the line segment with endpoints (2, 3) and (6, 9) into a 3:2 ratio.
Answer: Coordinates = \( \left(\frac{2 \cdot 2 + 6 \cdot 3}{2 + 3}, \frac{2 \cdot 3 + 6 \cdot 9}{2 + 3}\right) = \left(\frac{22}{5}, \frac{33}{5}\right) \). - Question: Calculate the slope of the line passing through the points (5, 2) and (3, 8).
Answer: Slope = \( \frac{8 - 2}{3 - 5} = -3 \).
Chapter 3: Angles
- Question: Define complementary angles and provide an example.
Answer: Complementary angles are two angles whose measures add up to 90 degrees. Example: \( 30^\circ \) and \( 60^\circ \) are complementary. - Question: Explain the concept of vertical angles and provide an example.
Answer: Vertical angles are pairs of non-adjacent angles formed by intersecting lines. Example: \( \angle 1 \) and \( \angle 3 \) are vertical angles in the figure. - Question: Define supplementary angles and provide an example.
Answer: Supplementary angles are two angles whose measures add up to 180 degrees. Example: \( 120^\circ \) and \( 60^\circ \) are supplementary. - Question: Describe adjacent angles and provide an example.
Answer: Adjacent angles are two angles that share a common vertex and a common side, but do not overlap. Example: \( \angle ACD \) and \( \angle BCD \) in the figure. - Question: Define the term "angle bisector" and explain its significance.
Answer: An angle bisector is a ray that divides an angle into two congruent angles. It is significant because it divides the angle into equal parts.
Chapter 4: Parallels
- Question: Explain what it means for two lines to be parallel and provide a visual representation.
Answer: Two lines are parallel if they never intersect and are always the same distance apart. A visual representation would be railroad tracks. - Question: Describe alternate interior angles and provide an example.
Answer: Alternate interior angles are a pair of non-adjacent angles formed by a transversal intersecting two lines and are on opposite sides of the transversal. Example: \( \angle 3 \) and \( \angle 6 \) in the figure. - Question: Define corresponding angles and provide an example.
Answer: Corresponding angles are angles that occupy the same relative position at each intersection where a straight line crosses two others. Example: \( \angle 1 \) and \( \angle 5 \) in the figure. - Question: Explain the concept of interior angles and exterior angles of a triangle.
Answer: Interior angles are angles inside a triangle, while exterior angles are angles formed outside a triangle when one side is extended. - Question: State the theorem about the sum of interior angles of a triangle and provide a brief explanation.
Answer: The sum of the interior angles of a triangle is always \( 180^\circ \). This is because the three interior angles of a triangle form a straight line.
Chapter 5: Triangles and Congruence
- Question: Define the term "triangle" and describe its characteristics.
Answer: A triangle is a closed plane figure with three straight sides and three angles. It has a total of three vertices and three edges. - Question: Explain the concept of congruent triangles and provide an example.
Answer: Congruent triangles are triangles that have the same size and shape. Example: \( \triangle ABC \) and \( \triangle DEF \) are congruent if their corresponding sides and angles are equal. - Question: State and explain the criteria for proving two triangles congruent.
Answer: The criteria for proving two triangles congruent are Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL). - Question: Describe the concept of similarity of triangles.
Answer: Similar triangles are triangles that have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are proportional. - Question: Explain the significance of the Pythagorean Theorem in triangle congruence.
Answer: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It is often used to determine if triangles are congruent.
Chapter 6: More About Triangles
- Question: In triangle \(ABC\), angle \(A\) measures \(60^\circ\), and angle \(B\) measures \(70^\circ\). What is the measure of angle \(C\)?
Answer: Angle \(C\) measures \(50^\circ\) (Sum of angles in a triangle is \(180^\circ\)). - Question: Triangle \(ABC\) has sides of lengths \(5\), \(7\), and \(9\). Is triangle \(ABC\) acute, obtuse, or right?
Answer: Triangle \(ABC\) is obtuse (The square of the longest side \(9^2 = 81\) is greater than the sum of the squares of the other two sides \(5^2 + 7^2 = 25 + 49 = 74\)). - Question: In triangle \(ABC\), if \(AB = 8\), \(BC = 12\), and \(AC = 15\), what is the length of the altitude from vertex \(A\) to side \(BC\)?
Answer: The length of the altitude is \(9.6\) (Use Heron's formula to find the area of the triangle, then use the area formula \(A = \frac{1}{2}bh\) to find the altitude). - Question: Triangle \(ABC\) is equilateral with side length \(6\). Find the radius of the circumscribed circle.
Answer: The radius of the circumscribed circle is \(2\sqrt{3}\) (For an equilateral triangle, the radius of the circumscribed circle is equal to \( \frac{\sqrt{3}}{3} \times \text{side length}\)). - Question: Triangle \(ABC\) has side lengths of \(13\), \(14\), and \(15\). Find the area of triangle \(ABC\).
Answer: The area of triangle \(ABC\) is \(84\sqrt{6}\) (Use Heron's formula to find the area).
Chapter 7: Triangle Inequalities
- Question: In triangle \(ABC\), if \(AB = 7\), \(BC = 8\), and \(AC = 9\), is triangle \(ABC\) possible?
Answer: Yes, triangle \(ABC\) is possible (The sum of the lengths of any two sides of a triangle must be greater than the length of the third side). - Question: Triangle \(ABC\) has side lengths of \(4\), \(6\), and \(10\). Is triangle \(ABC\) acute, obtuse, or right?
Answer: Triangle \(ABC\) is obtuse (The square of the longest side \(10^2 = 100\) is greater than the sum of the squares of the other two sides \(4^2 + 6^2 = 16 + 36 = 52\)). - Question: In triangle \(ABC\), if \(AB = 5\), \(BC = 6\), and \(AC = x\), what is the possible range of values for \(x\)?
Answer: \(1 < x < 11\) (Using the triangle inequality theorem). - Question: Triangle \(ABC\) has side lengths of \(3\), \(4\), and \(6\). Determine the type of triangle formed by these side lengths.
Answer: Triangle \(ABC\) is scalene (All sides have different lengths). - Question: Triangle \(ABC\) has side lengths of \(2\), \(4\), and \(5\). Is triangle \(ABC\) possible?
Answer: No, triangle \(ABC\) is not possible (The sum of the lengths of any two sides of a triangle must be greater than the length of the third side).
Chapter 8: Quadrilaterals
- Question: In parallelogram \(ABCD\), if \(AB = 6\) and \(BC = 8\), what is the length of diagonal \(AC\)?
Answer: The length of diagonal \(AC\) is \(10\) (In a parallelogram, the diagonals bisect each other). - Question: In rhombus \(WXYZ\), if the length of one side is \(10\), what is the length of each diagonal?
Answer: Each diagonal has a length of \(10\) (In a rhombus, the diagonals are congruent). - Question: In trapezoid \(ABCD\), if \(AB = 5\), \(BC = 7\), and \(AD = 9\), what is the length of side \(CD\)?
Answer: The length of side \(CD\) is \(5\) (In an isosceles trapezoid, the non-parallel sides are congruent). - Question: In rectangle \(ABCD\), if \(AB = 8\) and \(AD = 15\), what is the length of diagonal \(AC\)?
Answer: The length of diagonal \(AC\) is \(17\) (In a rectangle, the diagonals are congruent). - Question: In square \(ABCD\), if \(AB = 10\), what is the length of each diagonal?
Answer: Each diagonal has a length of \(10\sqrt{2}\) (In a square, the diagonals are congruent and bisect each other at right angles).
Chapter 9: Proportions and Similarity
- Question: In triangle \(ABC\), if \(AB = 8\), \(BC = 12\), and \(AC = 18\), what is the length of the altitude from vertex \(A\) to side \(BC\)?
Answer: The length of the altitude is \(9\) (Using similar triangles). - Question: In similar triangles \(ABC\) and \(DEF\), if \(AB = 4\), \(BC = 6\), \(DE = 8\), and \(EF = 12\), what is the ratio of the perimeters of \(ABC\) to \(DEF\)?
Answer: The ratio of the perimeters is \( \frac{1}{2} \). - Question: In triangle \(ABC\), if \(AD\) is an altitude, \(AB = 12\), and \(AD = 9\), what is the length of \(AC\)?
Answer: The length of \(AC\) is \(15\) (Using similar triangles). - Question: In similar triangles \(ABC\) and \(DEF\), if \(AB = 6\), \(BC = 9\), \(DE = 8\), and \(EF = 12\), what is the ratio of the areas of \(ABC\) to \(DEF\)?
Answer: The ratio of the areas is \( \frac{4}{3} \). - Question: In triangle \(ABC\), if \(AB = 10\), \(AC = 15\), and \(BC = 20\), what is the length of the altitude from vertex \(A\) to side \(BC\)?
Answer: The length of the altitude is \(12\) (Using similar triangles).
Chapter 10: Polygons and Area
- Question: In regular hexagon \(ABCDEF\), if the side length is \(6\), what is the area of the hexagon?
Answer: The area of the hexagon is \(93\sqrt{3}\) (Use the formula for the area of a regular polygon). - Question: In parallelogram \(ABCD\), if \(AB = 8\) and the altitude is \(5\), what is the area of the parallelogram?
Answer: The area of the parallelogram is \(40\) (Area = base × height). - Question: In regular octagon \(ABCDEFGH\), if the apothem length is \(4\), what is the perimeter of the octagon?
Answer: The perimeter of the octagon is \(32 + 8\sqrt{2}\) (Use the formula for the perimeter of a regular polygon). - Question: In trapezoid \(ABCD\), if \(AB = 10\), \(CD = 15\), and the height is \(8\), what is the area of the trapezoid?
Answer: The area of the trapezoid is \(120\) (Area = \( \frac{1}{2} \) × sum of bases × height). - Question: In kite \(WXYZ\), if \(WX = 6\), \(WY = 8\), and \(XZ = 10\), what is the area of the kite?
Answer: The area of the kite is \(24\) (Area = \( \frac{1}{2} \) × product of diagonals).
Chapter 11: Circles
- Question: In a circle with radius \(5\), find the length of the arc intercepted by a central angle of \(120^\circ\).
Answer: The length of the arc is \(\frac{5}{3} \pi\). - Question: In a circle with diameter \(10\), find the length of the arc intercepted by an inscribed angle of \(45^\circ\).
Answer: The length of the arc is \(\frac{5}{2} \pi\). - Question: In a circle with radius \(7\), find the length of the arc intercepted by a central angle of \(150^\circ\).
Answer: The length of the arc is \(\frac{7}{2} \pi\). - Question: In a circle with circumference \(20\pi\), find the length of the arc intercepted by a central angle of \(60^\circ\).
Answer: The length of the arc is \(10\). - Question: In a circle with radius \(12\), find the area of the sector intercepted by a central angle of \(45^\circ\).
Answer: The area of the sector is \(3\pi\).
Chapter 12: Surface Area and Volume
- Question: Find the surface area of a cone with radius \(6\) and slant height \(10\).
Answer: The surface area of the cone is \(186\pi\). - Question: Find the volume of a cylinder with radius \(4\) and height \(12\).
Answer: The volume of the cylinder is \(192\pi\). - Question: Find the total surface area of a sphere with radius \(5\).
Answer: The total surface area of the sphere is \(100\pi\). - Question: Find the volume of a pyramid with base area \(36\) and height \(8\).
Answer: The volume of the pyramid is \(96\). - Question: Find the surface area of a cube with edge length \(10\).
Answer: The surface area of the cube is \(600\).
Chapter 13: Right Triangles and Trigonometry
- Question: In a right triangle, if the length of one leg is \(5\) and the length of the other leg is \(12\), find the length of the hypotenuse.
Answer: The length of the hypotenuse is \(13\). - Question: In a right triangle, if the length of one leg is \(8\) and the length of the hypotenuse is \(10\), find the measure of the acute angle opposite the leg of length \(8\).
Answer: The measure of the acute angle is \( \sin^{-1}\left(\frac{4}{5}\right) \). - Question: In a right triangle, if the measure of one acute angle is \(30^\circ\), find the ratio of the lengths of the side opposite the angle to the hypotenuse.
Answer: The ratio is \( \frac{1}{2} \). - Question: In a right triangle, if the length of one leg is \(3\) and the length of the other leg is \(4\), find the measure of the acute angle opposite the leg of length \(3\).
Answer: The measure of the acute angle is \( \tan^{-1}\left(\frac{3}{4}\right) \). - Question: In a right triangle, if the length of one leg is \(6\) and the length of the other leg is \(8\), find the length of the hypotenuse.
Answer: The length of the hypotenuse is \(10\).
Chapter 14: Circle Relationships
- Question: In a circle with radius \(9\), find the length of a tangent segment from a point \(10\) units away from the center of the circle.
Answer: The length of the tangent segment is \( \sqrt{19} \). - Question: In a circle with radius \(6\), find the length of a secant segment if the external segment length is \(8\).
Answer: The length of the secant segment is \(10\). - Question: In a circle with diameter \(10\), find the length of a chord \(6\) units away from the center of the circle.
Answer: The length of the chord is \(8\). - Question: In a circle with radius \(5\), find the length of a tangent segment from a point \(13\) units away from the center of the circle.
Answer: The length of the tangent segment is \(12\). - Question: In a circle with radius \(12\), find the length of a chord \(16\) units away from the center of the circle.
Answer: The length of the chord is \(16\sqrt{3}\).
Chapter 15: Formalizing Proof
- Question: Prove that the diagonals of a rectangle are congruent.
Answer: Let \(ABCD\) be a rectangle. Diagonals \(AC\) and \(BD\) intersect at \(O\). In triangles \(AOC\) and \(DOB\), \(AO = DO\) (opposite sides of a rectangle are equal) and \(OC = OB\) (opposite sides of a rectangle are equal). Therefore, \(AOC \cong DOB\) (SSS congruence criterion). Hence, \(AC = BD\). - Question: Prove that the opposite sides of a parallelogram are congruent.
Answer: Let \(ABCD\) be a parallelogram. Opposite sides are parallel, so \(AB \parallel CD\) and \(BC \parallel AD\). By alternate interior angles, \( \angle A = \angle D\) and \( \angle B = \angle C\). Therefore, triangles \(ABD\) and \(CBD\) are congruent by ASA criterion. Hence, \(AB = CD\) and \(AD = BC\). - Question: Prove that the base angles of an isosceles triangle are congruent.
Answer: Let \(ABC\) be an isosceles triangle with \(AB = AC\). Draw the altitude from \(A\) to \(BC\) at \(D\). \(AD\) bisects \(BC\) and is perpendicular to it. In triangles \(ABD\) and \(ACD\), \(AD\) is common, \(AB = AC\) (given), and \(BD = CD\) (perpendicular bisector theorem). Therefore, \(ABD \cong ACD\) by SAS criterion. Hence, \( \angle B = \angle C\). - Question: Prove that the opposite angles of a parallelogram are congruent.
Answer: Let \(ABCD\) be a parallelogram. \(AB \parallel CD\) and \(AD \parallel BC\). By corresponding angles, \( \angle A = \angle C\) and \( \angle B = \angle D\). Therefore, \(ABCD\) is a cyclic quadrilateral, and opposite angles are supplementary. Hence, \( \angle A = \angle C\) and \( \angle B = \angle D\). - Question: Prove that the sum of the angles of a triangle is \(180^\circ\).
Answer: Let \(ABC\) be a triangle. Extend side \(BC\) to \(D\), forming a straight line. \( \angle A + \angle BCD = 180^\circ\) (linear pair). But \( \angle BCD = \angle B + \angle C\) (exterior angle of triangle). Therefore, \( \angle A + \angle B + \angle C = 180^\circ\).
Chapter 16: More Coordinate Graphing and Transformations
- Question: Reflect the point \(A(3, 4)\) across the x-axis.
Answer: The reflected point is \(A'(3, -4)\). - Question: Reflect the point \(B(-5, 2)\) across the y-axis.
Answer: The reflected point is \(B'(5, 2)\). - Question: Reflect the point \(C(2, -7)\) across the origin.
Answer: The reflected point is \(C'(-2, 7)\). - Question: Rotate the point \(D(4, 3)\) counterclockwise by \(90^\circ\) about the origin.
Answer: The rotated point is \(D'(-3, 4)\). - Question: Rotate the point \(E(-2, 6)\) clockwise by \(180^\circ\) about the origin.
Answer: The rotated point is \(E(2, -6)\).