a + b + c = ?
A hard-level "Length" problem.Count the same area two ways — split into three, and all at once. The two counts force the total, no matter where the point sits.
GivenThe equilateral triangle has height 15 cm. P is some point inside it.
Hints
- Join P to the three corners. The equilateral triangle splits into three small triangles.
- Look only at the blue triangle PBC. Its base is BC and its height is exactly a, so its area is BC × a ÷ 2.
- The other two work the same way. And since it is equilateral, all three sides are the same length — so the three triangles all share the same base length, differing only in height.
- Adding all three gives side × (a + b + c) ÷ 2, which is the area of the whole triangle.
- But the same area can be found the ordinary way: base × height ÷ 2. Same base, height 15 cm — so side × 15 ÷ 2.
- The two expressions describe the same area. Both have the form "side × □ ÷ 2", so the two □s must match: a + b + c = 15 cm.
- Nowhere did the argument use where P actually is. So wherever P moves inside the triangle, the three distances still total 15 cm.
Answer 15cm
More "Length" problems
- What is the side of the square?
- How long is MN?
- How long is the rope?
- How long is the pole's shadow?
- How deep was the water?
- How far does the centre travel?