Problem set (reference)
Class problems as we used them on this topic. For interactive practice with solutions and progress tracking, use Problems.cc when available.
1. Prove the following equalities:
a) (a + b)² = a² + 2ab + b²
b) (a − b)² = a² − 2ab + b²
c) (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc
d) a² − b² = (a + b)(a − b)
2. Given 4 right triangles with legs a, b and hypotenuse c, show that one can form a square of side a + b by adding:
a) one square of side c;
b) two squares of sides a and b.
3. Suppose a² + b² = c². Show how to dissect a square of side c into two squares of sides a and b (the number of parts should not depend on a and b).
4. In a rectangle drawn on squared paper, a diagonal is drawn. How many unit squares does it cross if the rectangle has size:
a) 7 × 13
b) 100 × 170
c) n × m
5. Suppose each matchstick has length 1 inch. Using 12 such matchsticks, form a figure with area 4 square inches.
6a. In triangle ABC, draw through A a line perpendicular to BC. Choose any point M on this line. Prove:
MB² − MC² = AB² − AC².
6b. In triangle ABC, suppose a point M in the plane satisfies
MB² − MC² = AB² − AC².
Prove that AM ⟂ BC.
7. From a point M inside triangle ABC, drop perpendiculars MP, MK, ME to sides AB, BC, CA respectively. Prove:
AP² + BK² + CE² = PB² + CK² + AE².
8. In an isosceles right triangle ABC, pick points M and K on hypotenuse AB (with K between M and B) so that ∠MCK = 45°. Prove that:
MK² = AM² + KB².
9. Find the locus of points from which the tangent lengths to two given circles are equal.