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The sum of any two even numbers is always an even number.

Let's denote two even numbers as 2n2n and 2m2m, where nn and mm are integers.

The sum of these two even numbers is: 2n+2m=2(n+m)2n+2m=2(n+m)

Since nn and mm are integers, n+mn+m is also an integer. Therefore, the sum 2(n+m)2(n+m) is an even number because it is divisible by 22. Thus, the sum of any two even numbers is an even number.

To construct a perpendicular to a given line segment at a point on it, you can follow these steps:

1. Draw the given line segment.
2. Choose a point on the line segment where you want the perpendicular to pass through. Let's call this point PP.
3. With point PP as the center, draw a circle with any radius that intersects the line segment at two points, let's call them AA and BB.
4. Draw a straight line through points AA and BB. This line will be perpendicular to the given line segment at point PP.

Now, you have constructed a perpendicular to the given line segment at the chosen point PP.

The greatest three-digit number that remains the same when its digits are reversed is also a palindrome.

The greatest three-digit palindromic number is 999.

When you reverse the digits of 999, you still get 999. So, it is the greatest three-digit number that does not change when its digits are reversed.

To get a number whose digits are the reverse of 203, we need to find the difference between this number and its reverse.

The number 203, when its digits are reversed, becomes 302.

To find what must be added to 203 to get 302, we subtract 203 from 302:

302−203=99302−203=99

So, 99 must be added to 203 to get a number whose digits are the reverse of the given number.

To divide 60 in the ratio of 2 : 3, we need to find two parts such that their ratio is 2 : 3.

Let the parts be 2x and 3x, where x is the common factor.

According to the given ratio: 2x+3x=602x+3x=60

Combine like terms: 5x=605x=60

Now, solve for x: x=605x=560 x=12x=12

Now, we can find the values of the parts: Part 1=2x=2×12=24Part 1=2x=2×12=24 Part 2=3x=3×12=36Part 2=3x=3×12=36

So, the parts in the ratio of 2 : 3 are 24 and 36, respectively.

To find the ratio of money withdrawn to the total money deposited, we simply divide the amount withdrawn by the total amount deposited.

Given:

• Amount deposited = ₹2050
• Amount withdrawn = ₹410

Now, we'll calculate the ratio of money withdrawn to the total money deposited: Ratio=Amount withdrawn/Total amount deposited

Substitute the given values: Ratio=410/2050

Now, simplify the fraction: Ratio=410/2050=41/205

So, the ratio of money withdrawn to the total money deposited is 41 : 205.

In this equilateral triangle ABC, AD is the median drawn from vertex A to the midpoint of BC (denoted as D), and AM is the altitude drawn from vertex A perpendicular to BC (denoted as M).

In an equilateral triangle, all medians are also altitudes. So, AD and AM coincide in this case.

Sure, let's draw five triangles and measure their sides. Then, we'll check if the sum of the lengths of any two sides is always less than the third side. For simplicity, I'll denote the sides of each triangle as aa, bb, and cc.

1. Triangle 1: Let's say a=3a=3 cm, b=4b=4 cm, c=5c=5 cm (a Pythagorean triplet). Checking the triangle inequality: a+b>ca+b>c is true, a+c>ba+c>b is true, and b+c>ab+c>a is true. So, this triangle follows the triangle inequality.

2. Triangle 2: Let's say a=5a=5 cm, b=6b=6 cm, c=2c=2 cm. Checking the triangle inequality: a+b>ca+b>c is true, a+c>ba+c>b is true, and b+c>ab+c>a is false (5 + 2 is not greater than 6). So, this triangle does not follow the triangle inequality.

3. Triangle 3: Let's say a=7a=7 cm, b=2b=2 cm, c=9c=9 cm. Checking the triangle inequality: a+b>ca+b>c is true, a+c>ba+c>b is true, and b+c>ab+c>a is true. So, this triangle follows the triangle inequality.

4. Triangle 4: Let's say a=4a=4 cm, b=10b=10 cm, c=5c=5 cm. Checking the triangle inequality: a+b>ca+b>c is true, a+c>ba+c>b is false (4 + 5 is not greater than 10), and b+c>ab+c>a is true. So, this triangle does not follow the triangle inequality.

5. Triangle 5: Let's say a=8a=8 cm, b=2b=2 cm, c=4c=4 cm. Checking the triangle inequality: a+b>ca+b>c is true, a+c>ba+c>b is true, and b+c>ab+c>a is false (2 + 4 is not greater than 8). So, this triangle does not follow the triangle inequality.

Based on the measurements and checks, not all the triangles follow the triangle inequality theorem. Specifically, triangles 2, 4, and 5 violate the inequality.

To find the fraction of a clockwise revolution that the hour hand of a clock turns through when it goes from 3 to 9, we first need to determine the total angle turned by the hour hand in a clockwise direction.

The total angle of a clock is 360 degrees. The hour hand moves through 12 hours to complete one full revolution.

From 3 to 9, the hour hand moves through 6 hours.

So, the fraction of a clockwise revolution turned through by the hour hand from 3 to 9 is:

Fraction=Number of hours passed/Total hours in a clock=6/12=1/2

Therefore, the hour hand turns through 1221 of a clockwise revolution when it goes from 3 to 9 on a clock.

The letter "Z" of the English alphabet has 2 lines of symmetry.

1. The first line of symmetry is vertical, passing through the center of the letter, dividing it into two mirror-image halves.
2. The second line of symmetry is diagonal, passing from the upper-left corner to the lower-right corner, dividing the letter into two congruent triangles.

So, the letter "Z" has 2 lines of symmetry.