Section formula

Key takeaways

Section formula belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Section formula to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Section formula from memory before moving on to harder problems.

Reference excerpt

In coordinate geometry, the Section formula is a formula used to find the ratio in which a line segment is divided by a point internally or externally. It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc.

Internal divisions

If point P (lying on AB) divides the line segment AB joining the points A ( x 1 , y 1 ) {\displaystyle \mathrm {A} (x_{1},y_{1})} and B ( x 2 , y 2 ) {\displaystyle \mathrm {B} (x_{2},y_{2})} in the ratio m:n, then

P = ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) {\displaystyle P=\left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}}\right)}

The ratio m:n can also be written as m / n : 1 {\displaystyle m/n:1} , or k : 1 {\displaystyle k:1} , where k = m / n {\displaystyle k=m/n} . So, the coordinates of point P {\displaystyle P} dividing the line segment joining the points A ( x 1 , y 1 ) {\displaystyle \mathrm {A} (x_{1},y_{1})} and B ( x 2 , y 2 ) {\displaystyle \mathrm {B} (x_{2},y_{2})} are:

( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) {\displaystyle \left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}}\right)}

= ( m n x 2 + x 1 m n + 1 , m n y 2 + y 1 m n + 1 ) {\displaystyle =\left({\frac {{\frac {m}{n}}x_{2}+x_{1}}{{\frac {m}{n}}+1}},{\frac {{\frac {m}{n}}y_{2}+y_{1}}{{\frac {m}{n}}+1}}\right)}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Section formula

Start with the simplest possible case. Write down what Section formula claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Section formula before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Section formula ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Section formula

In research: Section formula appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Section formula in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Section formula is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analytic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Section formula outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Section formula in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Section formula means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Section formula out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Section formula in simple terms?

In coordinate geometry, the Section formula is a formula used to find the ratio in which a line segment is divided by a point internally or externally. It is used to find out the centroid, incenter and excenters of a triangle.

Why does Section formula matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Section formula?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Section formula.