The Section (Midpoint) Formula

The Midpoint's x-coordinate

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The midpoint formula for the segment from \(P_1(x_1, y_1)\) to \(P_2(x_2, y_2)\) gives the x-coordinate of the midpoint as:

(x1 − x2)/2

(x1 + x2)/2

x1 + x2

(x1 · x2)/2

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Find a Midpoint

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Find the midpoint of the segment joining (2, 4) and (8, 10). The midpoint is ((1), (2)).

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5

7

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The Congruent Triangles in the Derivation

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Missing

Figure used to derive the midpoint formula

In the derivation of the midpoint formula, triangles P₁RM and MTP₂ are shown to be ______.

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The General Section Formula

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The general section formula for a point \(P(x, y)\) dividing the segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) internally in the ratio \(m:n\) gives \(x =\)

(m·x1 + n·x2)/(m − n)

(n·x2 + m·x1)/(m + n)

(m·x2 + n·x1)/(m + n)

(m·x2 − n·x1)/(m + n)

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Section Formula Reduces to the Midpoint Formula

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The section formula for internal division in the ratio m:n reduces to the ordinary midpoint formula when:

n = 0

m > n

m = n

m = 0

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Divide a Segment in a Given Ratio

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A point P divides the segment from A(0, 0) to B(10, 0) internally in the ratio 3:2. Find the x-coordinate of P.

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Order the Steps of the Derivation

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Arrange the steps used to derive the midpoint formula in the correct logical order.

Solve x − x1 = x2 − x and y − y1 = y2 − y

Note that triangles P1RM and MTP2 are congruent

Obtain x = (x1+x2)/2 and y = (y1+y2)/2

Set |P1R| = |MT| and |RM| = |TP2|

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The 1:1 Ratio Case

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In the section formula \(x = \dfrac{m\cdot x_2 + n\cdot x_1}{m+n}\), if \(m:n = 1:1\), this formula becomes:

x = (x1 + x2)/2

x = x1 + x2

x = x1 − x2

x = (x1·x2)/2

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Assumptions in the Derivation

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In deriving the section formula \(x=\frac{mx_2+nx_1}{m+n},\ y=\frac{my_2+ny_1}{m+n}\), which assumption was made about the coordinates?

x1 = x2

x2 < x1 and y2 < y1

No assumption is needed and the formula never holds otherwise

x2 > x1 and y2 > y1

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The Midpoint's x-coordinate

Correct Answer

Explanation

Find a Midpoint

Correct Answer

Explanation

The Congruent Triangles in the Derivation

Correct Answer

Explanation

The General Section Formula

Correct Answer

Explanation

Section Formula Reduces to the Midpoint Formula

Correct Answer

Explanation

Divide a Segment in a Given Ratio

Correct Answer

Explanation

Order the Steps of the Derivation

Correct Answer

Explanation

The 1:1 Ratio Case

Correct Answer

Explanation

Assumptions in the Derivation

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Explanation

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