Learn Maths with Free Lessons & Tips

To find the factors of a number, you need to identify all the numbers that divide the given number evenly, including 1 and the number itself.

Let's find the factors for each:

(a) Factors of 12: 1, 2, 3, 4, 6, 12

(b) Factors of 18: 1, 2, 3, 6, 9, 18

These are all the possible factors of 12 and 18.

To draw an angle of 120° and then construct an angle of 105°, follow these steps:

1. Draw an Angle of 120°:

   • Use a protractor to draw a straight line (ray) and mark a point on it to represent the vertex of the angle.
   • Place the center of the protractor at the vertex and align the baseline of the protractor with the ray.
   • Mark a point at the 120° mark on the protractor.
   • Draw a line from the vertex to the marked point to complete the angle.

2. Construct an Angle of 105°:

   • Place the center of the protractor at the vertex of the 120° angle.
   • Align the baseline of the protractor with one side of the 120° angle.
   • Mark a point at the 105° mark on the protractor.
   • Draw a line from the vertex to the marked point to construct the 105° angle.

Now, you have both the 120° angle and the constructed 105° angle.

To draw triangle ABC and its perpendiculars from each vertex onto the opposite sides, follow these steps:

1. Draw triangle ABC:

   • Draw three non-parallel lines intersecting at points A, B, and C to form triangle ABC.

2. Draw perpendiculars from each vertex onto the opposite sides:

   • From vertex A, draw a line perpendicular to side BC. Label the point of intersection with side BC as D.
   • From vertex B, draw a line perpendicular to side AC. Label the point of intersection with side AC as E.
   • From vertex C, draw a line perpendicular to side AB. Label the point of intersection with side AB as F.

Now, you have triangle ABC with perpendiculars AD, BE, and CF drawn from vertices A, B, and C respectively onto the opposite sides BC, AC, and AB.

To determine if these perpendiculars are concurrent (passing through the same point), you can construct the altitudes of the triangle, which are known to be concurrent at the orthocenter.

If the perpendiculars AD, BE, and CF intersect at a single point, then they are concurrent, and that point is the orthocenter of triangle ABC. If they do not intersect at a single point, then they are not concurrent.

Please note that the concurrency of perpendiculars in a triangle depends on the specific properties of the triangle. In some cases, the perpendiculars may be concurrent, while in others, they may not be. You would need to verify this for the specific triangle you've drawn.

The greatest 6-digit number is 999,999.

To find its successor, we simply add 1 to it:

999,999+1=1,000,000999,999+1=1,000,000

So, the successor of the greatest 6-digit number, 999,999, is 1,000,000.

To find the third term in the proportion when the first, second, and fourth terms are given, we can use the formula for proportions.

In a proportion, the product of the means is equal to the product of the extremes. Therefore, we have:

32/112=x/217

Where x is the third term that we need to find.

To solve for x, we can cross multiply:

32×217=112×x32×

6944=112x

Now, divide both sides by 112:

x=6944/112

x=62

So, the third term in the proportion is 62.

Through any two distinct points, exactly one line can be drawn. This is a fundamental postulate in geometry, known as the "line through two points" postulate. So, if you have two points, there is a unique line that passes through both of them.

The least number of line segments required to make a polygon is 3.

A polygon is a closed shape formed by straight lines. The simplest polygon is a triangle, which has three sides (and therefore three line segments).

So, with three line segments, you can form the smallest polygon, which is a triangle.

To find the number of right angles turned through by the hour hand of a clock when it goes from 3 to 6, we need to calculate the angle turned by the hour hand during this period and then divide it by the measure of a right angle, which is 90 degrees.

At 3 o'clock, the hour hand points directly at 3, which is 90 degrees clockwise from 12.

At 6 o'clock, the hour hand points directly at 6, which is 180 degrees clockwise from 12.

So, the angle turned by the hour hand from 3 to 6 is 180∘−90∘=90∘180∘−90∘=90∘.

Now, to find the number of right angles turned through, we divide this angle by the measure of a right angle:

Number of right angles=90∘90∘=1

Therefore, the hour hand of the clock turns through 1 right angle when it goes from 3 to 6.

If you start facing south and turn clockwise to west, you make a 90-degree clockwise turn to the west.

Since a right angle measures 90 degrees, you make exactly one right angle turn during this movement from south to west.

So, you make 1 right angle when you start facing south and turn clockwise to west.

A regular hexagon has 6 sides.

Since 6 is an even number, we can use the formula mentioned earlier: Number of lines of symmetry = n/2n/2

For a regular hexagon, with n=6n=6: Number of lines of symmetry = 6/2=36/2=3

So, a regular hexagon has 3 lines of symmetry.