Geometry problem solving: given, claim and plan
Separate the given from the claim, plan before writing a proof, find the shortest road with a perpendicular and Pythagoras, and see circle applications.
- Iran: grade 9
Guide
To solve a geometry problem: read it and mark the given and the claim; draw a suitable figure and mark what is given; plan which theorem or congruence case reaches the claim; then write the proof with reasons.
In isosceles triangle ABC with M the midpoint of BC, proving △ABM ≅ △ACM makes the corresponding parts equal. That is a plan: first find the two triangles, then choose the congruence case.
The shortest path from a point to a line is the perpendicular. Any other path is the hypotenuse of a right-angled triangle, found with Pythagoras' theorem.
In a circle the tangent is perpendicular to the radius at the point of contact, so we use Pythagoras in the triangle centre, point of contact, outside point. Two tangents from one point are equal, and the median to the hypotenuse of a right triangle is half the hypotenuse, since the right angle lies on a circle with the hypotenuse as diameter.