Point

Line

Planes

Find the distance between the points (1,2) and (4,6).

Step 1: Identify the coordinates of the points.

(x1​,y1​)=(1,2)

(x2​,y2​)=(4,6)

Step 2: Substitute the coordinates into the distance formula:

(picture above)

So, the distance between (1,2) and (4,6) is 5 units.

Example:

Find the midpoint between the points (1,2) and (4,6).

Step 1: Identify the coordinates of the points.

(x1​,y1​)=(1,2)

(x2​,y2​)=(4,6)

Step 2: Substitute the coordinates into the midpoint formula:

(picture above)

So, the midpoint between (1,2) and (4,6) is (2.5, 4).

Components of a conditional statement:

Hypothesis (P): The "if" part of the statement.

Conclusion (Q): The "then" part of the statement.

The truth value of a conditional statement is true unless the hypothesis is true, and the conclusion is false. If the hypothesis is true and the conclusion is false, then the statement is false.

A Venn diagram is a diagram used to show the relationships between different things. In geometry, venn diagrams are useful for showing us relationships between multiple geometric concepts or sets, like shapes, angles, or points.

Set: Points, angles, or figures.

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In a coordinate plane, if two lines have equations of the form:

Line 1: y=mx+b

Line 2: y=mx+b

The lines are parallel if m1=m2, where m represents the slope.

Perpendicular lines are lines that intersect at a right angle (90 degrees).

Two lines are perpendicular if the product of their slopes is −1.

In a coordinate plane, if the equations of two lines are:

Line 1: y=mx+b

Line 2: y=mx+b

The lines are perpendicular if m1 ⋅ m2=−1.

A point on the line, (x1,y1)(x,1, y,1)

The slope of the line, m.

The point-slope form of a line is given by the equation:

y−y1=m(x−x1)

Where:

(x1,y1)(x,1, y,1) is a point on the line, m is the slope of the line.

The standard form of the equation of a line is written as:

Ax+By=C

What it means: If all three sides of one triangle are the same length as the sides of another triangle, the triangles are congruent (same size and shape).

2. SAS (Side-Angle-Side)

What it means: If two sides and the angle between them in one triangle are the same as in another triangle, the triangles are congruent.

3. ASA (Angle-Side-Angle)

What it means: If two angles and the side between them are the same in both triangles, the triangles are congruent.

4. AAS (Angle-Angle-Side)

What it means: If two angles and a side (not between the angles) are the same in both triangles, the triangles are congruent.

5. HL (Hypotenuse-Leg) (Only for right triangles)

What it means: In right triangles, if the hypotenuse (the longest side) and one leg (a shorter side) are the same in both triangles, the triangles are congruent.

Steps to identify a rectangle:

A rectangle always has four right angles.

The opposite sides should be the same length.

The opposite sides of the rectangle should go in the same direction, meaning they are parallel to each other.

Example:

Imagine a rectangle with sides 4 cm and 6 cm.

All four angles are 90°.

Opposite sides (4 cm and 6 cm) are equal in length.

Opposite sides are parallel.

This shape is a rectangle.

A rhombus has all sides the same length.

Just like any parallelogram, opposite sides are parallel.

The opposite angles of a rhombus are the same.

The diagonals of a rhombus cut each other in half at 90° angles.

Example:

Imagine a rhombus with all sides measuring 5 cm.

All sides are the same length (5 cm).

Opposite sides are parallel.

Opposite angles are equal.

The diagonals bisect each other at 90°.

This shape is a rhombus.

Steps to Identify a Square:

A square has all sides the same length.

A square has four right angles (90°).

The diagonals in a square are the same length.

The diagonals intersect at 90° (right angle).

Example:

• Imagine a square with sides of 4 cm.

  All sides are equal (4 cm).

  All angles are 90°.

  The diagonals are equal in length and intersect at 90°.

  This shape is a square.

Steps to Identify a Trapezoid:

A trapezoid has only one pair of opposite sides that are parallel.

The other two sides (legs) are not parallel to each other.

The legs can be of different lengths and angles, but they will not be parallel.

Example:

• Imagine a trapezoid with bases of 6 cm and 10 cm, and legs of 4 cm and 5 cm.

  The two opposite sides (6 cm and 10 cm) are parallel.

  The other two sides (4 cm and 5 cm) are not parallel.

  This shape is a trapezoid.

These are the two sides of the trapezoid that are parallel to each other.

The midsegment connects the midpoints of the legs.

The length of the midsegment is the average of the lengths of the two bases. (Picture above)

A kite has two pairs of sides that are equal, and each pair is adjacent.

The diagonals of a kite intersect each other at 90°.

The diagonal that connects the unequal angles (the longer diagonal) bisects the other diagonal into two equal parts.

Example:

• Imagine a kite with two equal sides of 6 cm and two equal sides of 8 cm.

  The adjacent sides are equal (6 cm and 6 cm, 8 cm and 8 cm).

  The diagonals intersect at a right angle (90°).

  The longer diagonal bisects the shorter diagonal.

  This shape is a kite.

2nd slide: https://www.chilimath.com/lessons/intermediate-algebra/distance-formula/

3rd slide:https://www.youtube.com/watch?v=pzDfd8NXRXk

4th slide: https://slideplayer.com/slide/9750849/

5th slide: https://sciencenotes.org/parallel-and-perpendicular-lines/

6th slide: https://knowunity.com/knows/algebra-1-linear-equations-06a0b255-ed4d-468b-9cec-18c9b46619cc

7th slide: https://knowunity.com/knows/algebra-1-linear-equations-06a0b255-ed4d-468b-9cec-18c9b46619cc

8th slide: https://www.cuemath.com/geometry/parallelograms/

9th slide: https://www.youtube.com/watch?v=Ov4aTty67A8

10th slide: https://byjus.com/maths/rhombus/

11th slide: https://www.cuemath.com/geometry/square/

12th slide: https://byjus.com/maths/trapezoids/

13th slide: https://andymath.com/trapezoid-midsegment/

14th slide:https://www.youtube.com/watch?v=t_hl06jGIIQ