Coordinate Geometry: Midsegments and Parallel Lines

11th Grade

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| Attempts: 12 | Questions: 20 | Updated: Jan 16, 2026

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1) In △ABC, A(0,0), B(6,0), C(0,8). Midpoints of AB and AC are M(3,0) and N(0,4). What is MN's equation?

Y = –(4/3)x + 4

Y = 0

Y = –(2/3)x + 4

Y = 0.5x + 2

Step 1: Find the slope of MN → m=(4−0)/(0−3)=−4/3m = (4 - 0) / (0 - 3) = -4/3m=(4−0)/(0−3)=−4/3.

Step 2: Using point (0, 4), the equation becomes y=−43x+4y = -\frac{4}{3}x + 4y=−34x+4.

So, MN’s equation is y = –(4/3)x + 4.

Explanation

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About This Quiz

Coordinate Geometry: Midsegments and Parallel Lines - Quiz

Get ready to use points on a grid to explore midsegments! In this quiz, you’ll find midpoints, use slopes, and show that a midsegment is parallel to a side and half its length. You’ll practice writing equations of lines and measuring distances, all with simple steps. By the end, coordinates... see morewill help you prove midsegment facts with confidence.
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2)

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2) For the same triangle, BC's equation is y = –(4/3)x + 8. Is MN parallel to BC?

Yes, slope of MN = slope of BC.

No, slopes differ.

Yes, slopes are negative reciprocals.

No, MN is vertical.

Both lines (MN and BC) have slope –4/3.

Since parallel lines have the same slope, MN ∥ BC.

Explanation

Both lines (MN and BC) have slope –4/3.

Since parallel lines have the same slope, MN ∥ BC.

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3) The midsegment drawn parallel to the longest side of a triangle will always be:

Parallel and half the length of that side

Perpendicular to that side

Equal in length to that side

Twice as long as that side

By the Midsegment Theorem, the midsegment is parallel to one side and half as long.

Explanation

By the Midsegment Theorem, the midsegment is parallel to one side and half as long.

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4) If BC's slope is 2, the slope of the midsegment MN is:

–½

Undefined

A midsegment is parallel to the base → it has the same slope, 2.

Explanation

A midsegment is parallel to the base → it has the same slope, 2.

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5) The midpoint formula is:

((x₁–x₂)/2, (y₁–y₂)/2

((x₁+x₂)/2, (y₁+y₂)/2)

((x₁+x₂)/3, (y₁+y₂)/3)

((x₁+x₂)/4, (y₁+y₂)/4)

The midpoint formula calculates the point that is exactly halfway between two given points (x₁, y₁) and (x₂, y₂) in a Cartesian coordinate system. It averages the x-coordinates and the y-coordinates of the points to find the midpoint.

Explanation

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6) The midsegment rule is used in coordinate proofs to show:

Perpendicularity

Congruence

Symmetry

Parallelism and proportionality

Using slopes (for parallel lines) and distances (for proportionality), midsegments prove both parallel and half-length relationships.

Explanation

Using slopes (for parallel lines) and distances (for proportionality), midsegments prove both parallel and half-length relationships.

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7) Which is true if MN ∥ QR?

Slope of MN = slope of QR

Slope of MN × slope of QR = –1

Slope of MN = 0

Slope of MN is undefined

If two lines are perpendicular (MN ⊥ QR), the product of their slopes is -1. This indicates that they intersect at a right angle.

Explanation

If two lines are perpendicular (MN ⊥ QR), the product of their slopes is -1. This indicates that they intersect at a right angle.

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8) If BC = √((x₂–x₃)²+(y₂–y₃)²), then MN =

MN = BC

MN = ½BC

MN = 2BC

MN = BC²

In this case, MN is defined as half the length of BC, which is why MN = ½BC is the correct expression.

Explanation

In this case, MN is defined as half the length of BC, which is why MN = ½BC is the correct expression.

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9) Which geometric property is always used in midsegment proofs?

Parallel lines

Perpendicular bisectors

Angle bisectors

Radii of circles

In midsegment proofs, the midsegment is always parallel to the third side of the triangle, which is why the property of parallel lines is essential.

Explanation

In midsegment proofs, the midsegment is always parallel to the third side of the triangle, which is why the property of parallel lines is essential.

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10) The slope of a midsegment equals the slope of:

The median

The base it’s parallel to

The altitude

The angle bisector

A midsegment is parallel to the base, which means it has the same slope as that side.

Explanation

A midsegment is parallel to the base, which means it has the same slope as that side.

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11) In triangle ABC, M and N are the midpoints of AB and AC. If M(1,6) and N(−1,2), what is the slope of MN?

–2

–½

Slope = (2 – 6) / (–1 – 1) = (–4) / (–2) = 2.

But direction from M to N gives –2, so slope = –2.

Explanation

Slope = (2 – 6) / (–1 – 1) = (–4) / (–2) = 2.

But direction from M to N gives –2, so slope = –2.

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12) If a triangle's vertices are (0,0), (6,0), (0,8), the midsegment connecting midpoints of AB and AC has length:

The midsegment of a triangle connects the midpoints of two sides and is parallel to the third side. The length of the midsegment is half the length of the third side. In this case, the length of side BC is 10, so the length of the midsegment is 10 / 2 = 5.

Explanation

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13) In △PQR, P(0,0), Q(4,0), R(0,6). Midpoints M(2,0) and N(0,3). MN's length = ?

√13

√10

To find the length of segment MN, we can use the distance formula. The coordinates of M are (2,0) and the coordinates of N are (0,3). The distance D between two points (x1, y1) and (x2, y2) is given by the formula: D = √((x2 - x1)² + (y2 - y1)²). Substituting the coordinates of M and N, we get: D = √((0 - 2)² + (3 - 0)²) = √(4 + 9) = √13. Thus, the correct length of MN is √13.

Explanation

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14) If a triangle has vertices (2,2), (6,2), (4,8), midpoints of the legs are (4,2) and (3,5). The midsegment's slope is:

–3

To calculate the slope of the midsegment, use the coordinates of its endpoints, which are the midpoints of the two legs of the triangle. The formula for slope (m) is m = (y2 - y1) / (x2 - x1). Here, using (4,2) and (3,5), we find the slope to be (5 - 2) / (3 - 4) = 3 / -1 = -3.

Explanation

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15) Midsegments prove similarity because:

They divide sides in a 1:2 ratio.

They form smaller similar triangles.

They bisect angles.

They form parallelograms.

The smaller triangle formed by connecting midpoints has sides parallel and proportional to the original, so they’re similar.

Explanation

The smaller triangle formed by connecting midpoints has sides parallel and proportional to the original, so they’re similar.

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16) The distance formula proves midsegments are:

Half the base

Equal to the base

Perpendicular to the base

Longer than the base

The distance formula shows that the length of the midsegment in a triangle is always half the length of the base it is parallel to, adhering to the properties of similar triangles.

Explanation

The distance formula shows that the length of the midsegment in a triangle is always half the length of the base it is parallel to, adhering to the properties of similar triangles.

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17) Which method is often used in proofs involving midsegments?

Law of Cosines

Slope and Midpoint Formulas

Trigonometric Ratios

Law of Sines

The Slope and Midpoint Formulas are commonly used to demonstrate properties of midsegments in triangles, such as showing that they are parallel to the third side and half its length.

Explanation

The Slope and Midpoint Formulas are commonly used to demonstrate properties of midsegments in triangles, such as showing that they are parallel to the third side and half its length.

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18) A triangle's midsegments form a smaller triangle that is:

Congruent to the original

Isosceles

Equilateral

Similar to the original

The midsegments of a triangle connect the midpoints of two sides, forming a smaller triangle that is similar to the original triangle because its sides are proportional to the sides of the original triangle.

Explanation

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19) The midsegment theorem shows that △ formed inside is scaled by a factor of:

The midsegment theorem states that the segment connecting the midpoints of two sides of a triangle is half the length of the third side, hence the scaling factor is ½.

Explanation

The midsegment theorem states that the segment connecting the midpoints of two sides of a triangle is half the length of the third side, hence the scaling factor is ½.

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