Geometry

The One Thing

Geometry in Grade 10 establishes relationships among figures through transformations, proportional reasoning, right-triangle trigonometry, and coordinate methods. Accurate solutions depend on identifying corresponding parts, selecting an appropriate theorem or formula, showing logically justified work, and checking whether the result is reasonable.

Definitions and Results

• Congruent figures: Figures with the same shape and size; all corresponding sides and angles are equal. Congruence may be demonstrated by a sequence of rigid motions, including translations, rotations, reflections, or combinations of these transformations.
• Rigid motion: A transformation that preserves distances, angle measures, collinearity, and parallelism. A translation slides every point the same distance in the same direction; a rotation turns a figure around a fixed center by a specified angle; a reflection flips a figure across a line of reflection.
• Congruence criteria for triangles: Congruent triangles can be proved using SSS, SAS, ASA, AAS, or HL for right triangles. SSA is generally not a valid triangle congruence criterion. In the statement triangle ABC congruent to triangle DEF, the correspondence is A to D, B to E, and C to F.
• Similarity: A relationship between figures with equal corresponding angles and proportional corresponding sides. Similar figures have the same shape but may have different sizes.
• Scale factor: The ratio of a length in one figure to the corresponding length in a similar figure. Corresponding side lengths have a constant ratio called the scale factor. The perimeter ratio of similar figures equals the scale factor, while the area ratio equals the square of the scale factor.
• Similar triangles: Triangles with congruent corresponding angles and proportional corresponding side lengths.
• AA similarity: Two triangles are similar when two pairs of corresponding angles are congruent.
• SAS similarity: Two triangles are similar when two pairs of corresponding sides are proportional and the included angles are congruent.
• SSS similarity: Two triangles are similar when all three pairs of corresponding sides are proportional.
• Pythagorean theorem: In a right triangle, the squares of the leg lengths add to the square of the hypotenuse:

$$a^2+b^2=c^2.$$ The hypotenuse is opposite the right angle and is always the longest side. The theorem applies only to right triangles.

• Sine: For an acute angle $\theta$ in a right triangle,

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}.$$

• Cosine: For an acute angle $\theta$ in a right triangle,

$$\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}.$$

• Tangent: For an acute angle $\theta$ in a right triangle,

$$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}.$$

• Inverse trigonometric functions: ${\sin }^{-1}$, ${\cos }^{-1}$, and ${\tan }^{-1}$ are used to find an unknown angle from a trigonometric ratio. Calculators should be set to degree mode when angles are measured in degrees.
• Triangle angle results: The angles of a triangle sum to $180^{\circ }$. The acute angles of a right triangle are complementary.
• Slope: A measure of a line’s steepness, calculated by

$$m=\frac{y_2-y_1}{x_2-x_1}.$$ A horizontal line has slope $0$, while a vertical line has undefined slope.

• Slope-intercept form: The equation of a line may be written as

$$y=mx+b,$$ where $m$ is the slope and $b$ is the $y$-intercept.

• Parallel lines: Lines in the same plane that never intersect. When neither line is vertical, parallel lines have equal slopes.
• Perpendicular lines: Lines that intersect at a right angle. When both slopes are defined, their slopes are negative reciprocals. The equations

$$y=mx+b\text{and}y=-\frac{1}{m}x+c$$ represent perpendicular nonvertical lines when $m\ne 0$.

• Distance formula: Derived from the Pythagorean theorem, the distance between $(x_1,y_1)$ and $(x_2,y_2)$ is

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

• Midpoint: The point halfway between $(x_1,y_1)$ and $(x_2,y_2)$ is

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$ The midpoint formula can identify the center of a segment and prove that diagonals bisect each other.

• Coordinate proof: A proof using coordinates, algebra, slopes, distances, or midpoints to establish a geometric relationship.
• Coordinate transformations: A translation may be represented by

$$(x,y)\to (x+a,y+b).$$ A dilation centered at the origin with scale factor $k$ maps $$(x,y)\to (kx,ky).$$ A dilation preserves angle measures but changes lengths unless $k=1$.

Worked Methods

Proving congruence using transformations

• Identify the corresponding parts of the two figures.
• Apply translations, rotations, reflections, or a sequence of these rigid motions.
• Verify that the image coincides with the second figure.
• Conclude that the figures are congruent because rigid motions preserve distance, angle measure, collinearity, and parallelism.

Proving triangle congruence

• Match the vertices and sides in the correct correspondence.
• Identify the available equal sides and angles.
• Apply one of the valid criteria: SSS, SAS, ASA, AAS, or HL for right triangles.
• Do not use SSA as a generally valid triangle congruence criterion.
• State the congruence conclusion using the correct correspondence. For example, triangle ABC congruent to triangle DEF means A corresponds to D, B to E, and C to F.

Proving triangle similarity

• Identify corresponding angles and sides.
• Use AA when two pairs of corresponding angles are congruent.
• Use SAS when two pairs of corresponding sides are proportional and the included angles are congruent.
• Use SSS when all three pairs of corresponding sides are proportional.
• Determine the scale factor from corresponding side lengths.
• Apply the scale factor to other corresponding lengths. The perimeter ratio is the scale factor, and the area ratio is the square of the scale factor.

Applying the Pythagorean theorem

• Confirm that the triangle is a right triangle.
• Identify the hypotenuse as the side opposite the right angle and label it $c$.
• Label the legs $a$ and $b$.
• Substitute into

$$a^2+b^2=c^2.$$

• Solve for the unknown length.
• Check that the hypotenuse is the longest side.

Solving a right-triangle trigonometry problem

• Identify the given acute angle $\theta$.
• Relative to $\theta$, label the opposite side, adjacent side, and hypotenuse.
• Select the ratio:

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}},\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}},\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}.$$

• Substitute the known values and solve for the unknown side.
• To find an unknown angle, use ${\sin }^{-1}$, ${\cos }^{-1}$, or ${\tan }^{-1}$.
• Check that the calculator is in degree mode when the angle is measured in degrees.
• For a right triangle, use the fact that the two acute angles are complementary when appropriate.

Finding slope and equations of lines

• Substitute the coordinates of two points into

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Interpret the result: equal slopes indicate parallel nonvertical lines; negative reciprocal slopes indicate perpendicular lines.
• Use

$$y=mx+b$$ to write the equation of a line, where $m$ is the slope and $b$ is the $y$-intercept.

• Treat horizontal lines as having slope $0$ and vertical lines as having undefined slope.

Using the distance formula to classify a triangle

• Calculate the lengths of the three sides using

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

• Compare the lengths. Equal distances may show that the triangle is isosceles or equilateral.
• Compare the squared lengths. A Pythagorean relationship may show that the triangle is a right triangle.

Using the midpoint formula

• Identify the endpoints $(x_1,y_1)$ and $(x_2,y_2)$.
• Substitute them into

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• Use the resulting point to locate the center of a segment or to prove that diagonals bisect each other.

Writing a coordinate proof

• State the given information.
• Assign coordinates to the relevant points.
• Use slopes, distances, midpoints, equations, or coordinate rules.
• Apply the relevant definitions or theorems.
• Show the algebraic steps.
• Clearly justify the conclusion, such as parallelism, perpendicularity, symmetry, or congruence.

Where It Goes Wrong

• Confusing congruence with similarity: congruence preserves both shape and size, whereas similarity preserves shape but allows a proportional change in size.
• Misidentifying corresponding vertices, sides, or angles; the order in a congruence statement determines the correspondence.
• Applying the Pythagorean theorem to a triangle that is not right-angled, or failing to identify the hypotenuse as the side opposite the right angle.
• Choosing the wrong trigonometric ratio by mislabeling the opposite, adjacent, or hypotenuse side; an angle must be specified before these labels are assigned.
• Forgetting degree mode when using trigonometric functions with angles measured in degrees.
• Treating SSA as a valid general triangle congruence criterion, or overlooking that vertical lines have undefined slope and that perpendicular slopes are negative reciprocals only when both slopes are defined.

What Gets Asked

This material supports questions that require students to:

• Identify whether figures are congruent or similar and determine corresponding parts.
• Describe or apply translations, rotations, reflections, rigid-motion sequences, and coordinate transformation rules.
• Prove triangle congruence using SSS, SAS, ASA, AAS, or HL, and explain why SSA is generally insufficient.
• Prove triangle similarity using AA, SAS, or SSS and calculate scale factors, perimeter ratios, and area ratios.
• Use the Pythagorean theorem to find an unknown side in a right triangle and verify whether a triangle is right.
• Use sine, cosine, tangent, and inverse trigonometric functions to find unknown sides or angles in practical contexts such as surveying, construction, navigation, and measuring heights.
• Calculate slopes, write equations in the form $y=mx+b$, and determine whether lines are parallel or perpendicular.
• Find distances and midpoints from coordinates.
• Classify triangles using side lengths and the Pythagorean relationship.
• Use midpoint calculations to prove that diagonals bisect each other.
• Write coordinate proofs establishing parallelism, perpendicularity, symmetry, and congruence.
• Present a complete proof by stating given information, applying definitions or theorems, showing logical steps, and justifying the conclusion.