Question
Construct the angle with the help of ruler and compasses only:
90°
90°
✓
Answer
Draw a ray OA.
With a convenient radius and centre at O, draw an arc cutting the ray OA at P.
With the same radius and centre at P, draw another arc, which cuts the first arc at Q.
With the same radius and centre at Q, draw another arc, which cuts the first arc at R.
With Q and R as centres and radius more than half of QR, which cuts each other at S.
Draw OS and extend it to B from the ray OB.
$\angle\text{AOB}$ is required angle of 90°.
With a convenient radius and centre at O, draw an arc cutting the ray OA at P.
With the same radius and centre at P, draw another arc, which cuts the first arc at Q.
With the same radius and centre at Q, draw another arc, which cuts the first arc at R.
With Q and R as centres and radius more than half of QR, which cuts each other at S.
Draw OS and extend it to B from the ray OB.
$\angle\text{AOB}$ is required angle of 90°.
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Pravin, Bageshri and Yash are cousins who live in the same house. Pravin is an Army Officer. Bageshri is studying in a Medical College in another city. Yash lives in a nearby town in a hostel. Pravin can come home every 120 days.
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Yash said, “The number of days after which we come back together must be divisible by 30, 120. That means we shall be back together on the 10th of June next year. That will certainly be a for us!”
How did Yash find the answer?
Bageshri comes home every 45 days and Yash, every 30 days. All three of them left home at the same time on the 15th of June 2016. Their parents said, “We shall celebrate like a festival the day you all come home together.” Mother asked Yash, “What day will that be?”
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How did Yash find the answer?
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Kaprekar Number.
i. Take any 4-digit number in which all the digits are not the same.
ii. Obtain a new 4-digit number by arranging the digits in descending order.
iii. Obtain another 4-digit number by arranging the digits of the new number in ascending order.
iv. Subtract the smaller of these two new numbers from the bigger number. The difference obtained will be a 4-digit number. If it is a 3-digit number, put a 0 in the thousands place. Repeat the above steps with the difference obtained as a result of the subtraction.
v. After some repetitions, you will get the number 6174. If you continue to repeat the same steps you will get the number 6174 every time. Let us begin with the number 8531.
8531 → 7173 → 6354 → 3087 → 8352 → 6174 → 6174
This discovery was made by the mathematician, Dattatreya Ramchandra Kaprekar. That is why the number 6174 was named the Kaprekar number.
i. Take any 4-digit number in which all the digits are not the same.
ii. Obtain a new 4-digit number by arranging the digits in descending order.
iii. Obtain another 4-digit number by arranging the digits of the new number in ascending order.
iv. Subtract the smaller of these two new numbers from the bigger number. The difference obtained will be a 4-digit number. If it is a 3-digit number, put a 0 in the thousands place. Repeat the above steps with the difference obtained as a result of the subtraction.
v. After some repetitions, you will get the number 6174. If you continue to repeat the same steps you will get the number 6174 every time. Let us begin with the number 8531.
8531 → 7173 → 6354 → 3087 → 8352 → 6174 → 6174
This discovery was made by the mathematician, Dattatreya Ramchandra Kaprekar. That is why the number 6174 was named the Kaprekar number.
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6(1 - 4x) + 7(2 + 5x) = 53
6(1 - 4x) + 7(2 + 5x) = 53
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i. Collinear points
ii. Rays
iii. Line segments
iv. Lines
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