The Dancing Triangle and Its Applications

In the figure below, lines l and m are parallel lines. What can you say about the areas of triangle ABC and triangle ADC? The distance between two parallel lines is equal at any point, so the two triangles have the same altitude (can you see why?). Further, the two triangles have a common base, […]
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A US President’s Proof of the Pythagorean Theorem

James Garfield, the 20th president of the United States, came up with an original proof of the Pythagorean Theorem in 1876 when he was still a Congressman. His proof was published in New England Journal of Education. Recall that the Pythagorean Theorem states that given a right triangle with sides , , and hypotenuse , […]
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Area Tutorial 4 – Derivation of the Area of a Circle

We have learned that the area of a parallelogram is the product of its base and its height, and the circumference of a circle with radius is . To find the area of a circle with radius , divide it into congruent sectors (blue and red divisions) then arrange them as shown below. Figure 2 […]
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