Constructions, loci and geometric conditions · Higher · Construcciones, lugares geométricos y condiciones geométricas · Superior
| English | Español |
|---|---|
| locus/ˈləʊkəs/ | locus |
Where can both conditions hold?
- A router must be equally far from two rooms and within reach of a power point. Each condition creates a different set of possible positions.
- This lesson studies locus 轨迹: The set of all points satisfying a stated geometric condition.
Choose the mathematical structure
- Points equally distant from A and B lie on the perpendicular bisector of AB. Points at fixed distance r from C lie on a circle. Points equally distant from two intersecting lines lie on their angle bisectors.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines locus?
The set of all points satisfying a stated geometric condition.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For endpoints A and B 6 cm apart, draw equal-radius arcs above and below AB, with compass opening greater than 3 cm. Join the arc intersections to obtain the perpendicular bisector, crossing AB at its midpoint 3 cm from each end. To construct a perpendicular from P to a line, use a circle centred at P to mark two line points, then bisect their segment. For a perpendicular at P on the line, mark equal distances on each side of P and use equal arcs. For an angle bisector, draw one vertex-centred arc meeting both arms, then equal arcs from those two points; join their intersection to the vertex. Equal-radius circles centred at the endpoints of a segment construct an equilateral triangle and hence a 60° angle. For a point equally distant from A and B and 5 cm from A, intersect the bisector with a 5 cm circle centred at A. Each intersection is 4 cm perpendicular to AB by a 3–4–5 triangle. The shortest point-to-line distance follows a perpendicular, since any slanted route is a longer hypotenuse.
Constructions, loci and geometric conditions · Construcciones, lugares geométricos y condiciones geométricas
PA=PB
Match each geometric condition to its construction.
Find the midpoint x-coordinate between (0,0) and (6,0).
Midpoint x-coordinate=(0+6)/2=3.
Test a tempting shortcut
- The perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines. A sketch is not a ruler-and-compass construction: preserve arcs as evidence of the method.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Points equally distant from two points always lie on their angle bisector. This claim is false. Explain which definition or assumption it violates.
A locus intersection is 5 cm from A and its midpoint foot is 3 cm from A. Find its perpendicular height in cm.
The right triangle has hypotenuse 5 and horizontal leg 3: h²=5²-3²=16, so the positive height is 4.
Points equally distant from two points always lie on their angle bisector.
The perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines. A sketch is not a ruler-and-compass construction: preserve arcs as evidence of the method.
Interpret a new situation
- AQA G2 uses ruler-and-compass constructions, including a 60° angle, perpendiculars, bisectors and intersections of loci. Preserve construction arcs and justify the equidistance condition.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the distance from (3,4) to (0,0).
Distance=√(3²+4²)=5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The set of all points satisfying a stated geometric condition. Choose the relationship, show the method, check its assumptions and interpret the result.