Geometry, Plane Geometry, Pythagorean Theorem

The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (`a^2 + b^2 = c^2`). Questions involve applying this theorem to find side lengths.

  • STEALING THE BELL-ROPES

    Two men broke into a church tower one night to steal the bell-ropes. The two ropes passed through holes in the wooden ceiling high above them, and they lost no time in climbing to the top. Then one man drew his knife and cut the rope above his head, in consequence of which he fell to the floor and was badly injured. His fellow-thief called out that it served him right for being such a fool. He said that he should have done as he was doing, upon which he cut the rope below the place at which he held on. Then, to his dismay, he found that he was in no better plight, for, after hanging on as long as his strength lasted, he was compelled to let go and fall beside his comrade. Here they were both found the next morning with their limbs broken. How far did they fall? One of the ropes when they found it was just touching the floor, and when you pulled the end to the wall, keeping the rope taut, it touched a point just three inches above the floor, and the wall was four feet from the rope when it hung at rest. How long was the rope from floor to ceiling? Sources:

  • THE YORKSHIRE ESTATES

     

    I was on a visit to one of the large towns of Yorkshire. While walking to the railway station on the day of my departure a man thrust a hand-bill upon me, and I took this into the railway carriage and read it at my leisure. It informed me that three Yorkshire neighbouring estates were to be offered for sale. Each estate was square in shape, and they joined one another at their corners, just as shown in the diagram. Estate A contains exactly `370` acres, B contains `116` acres, and C `74` acres.

    Now, the little triangular bit of land enclosed by the three square estates was not offered for sale, and, for no reason in particular, I became curious as to the area of that piece. How many acres did it contain?

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  • FARMER WURZEL'S ESTATE

     

    I will now present another land problem. The demonstration of the answer that I shall give will, I think, be found both interesting and easy of comprehension.

    Farmer Wurzel owned the three square fields shown in the annexed plan, containing respectively `18, 20`, and `26` acres. In order to get a ring-fence round his property he bought the four intervening triangular fields. The puzzle is to discover what was then the whole area of his estate.

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