Triangle Midsegment Quiz: Master Triangle Midsegment Quiz

1) The Varignon Parallelogram is formed by connecting:

Altitudes

Medians

Midpoints of sides of a quadrilateral

Angle bisectors

Varignon’s theorem states that joining midpoints of any quadrilateral forms a parallelogram.

Explanation

Varignon’s theorem states that joining midpoints of any quadrilateral forms a parallelogram.

2) If the base of a triangle is 18 cm, the midsegment parallel to it measures __ cm.

Midsegment = ½ × 18 = 9 cm, directly following the midsegment theorem.

Explanation

Midsegment = ½ × 18 = 9 cm, directly following the midsegment theorem.

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3) The triangle midsegment theorem states that the midsegment is

Perpendicular to the third side

An angle bisector

Equal in length to the third side

Parallel to the third side and half its length

The midsegment theorem states that the segment connecting midpoints of two sides is parallel to the third side and half as long.

Explanation

The midsegment theorem states that the segment connecting midpoints of two sides is parallel to the third side and half as long.

4) A midsegment always connects the midpoints of two sides of a triangle.

True

False

By definition, a triangle midsegment connects the midpoints of two sides.

Explanation

By definition, a triangle midsegment connects the midpoints of two sides.

5) If DE is a midsegment in △ABC, and BC = 14 cm, then DE = __ cm.

The midsegment is half the base it’s parallel to, so DE = ½ × 14 = 7 cm.

Explanation

The midsegment is half the base it’s parallel to, so DE = ½ × 14 = 7 cm.

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6) If DE = 8 cm, then the parallel side BC measures:

4 cm

8 cm

12 cm

16 cm

Since DE is half the length of BC, BC = 2 × 8 = 16 cm.

Explanation

Since DE is half the length of BC, BC = 2 × 8 = 16 cm.

7) How many midsegments can be drawn in one triangle?

1

2

3

6

Each triangle has one midsegment for each pair of sides—there are three in total.

Explanation

Each triangle has one midsegment for each pair of sides—there are three in total.

8) If a triangle’s midsegment is 5 units, what is the length of the base it’s parallel to?

5

7.5

10

15

The base is always twice as long as its parallel midsegment, so 2 × 5 = 10 units.

Explanation

The base is always twice as long as its parallel midsegment, so 2 × 5 = 10 units.

9) Select all true properties of a midsegment.

Connects midpoints

Parallel to the base

Equal in length to the base

Half the length of the base

Forms a smaller similar triangle

Midsegments connect midpoints, are parallel to the third side, half as long, and form a smaller similar triangle.

Explanation

Midsegments connect midpoints, are parallel to the third side, half as long, and form a smaller similar triangle.

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10) Which theorem helps prove the triangle midsegment theorem?

Pythagorean Theorem

Similar Triangles

Triangle Inequality

Ceva’s Theorem

The proof relies on similar triangles because corresponding sides are proportional.

Explanation

The proof relies on similar triangles because corresponding sides are proportional.

11) Drawing all three midsegments in a triangle creates four smaller triangles of equal area.

True

False

The three midsegments divide the triangle into four smaller triangles, all congruent in area and shape.

Explanation

The three midsegments divide the triangle into four smaller triangles, all congruent in area and shape.

12) If the midsegment is parallel to the longest side, what can we say about their slopes in coordinate geometry?

Slopes are equal

Slopes are reciprocals

Slopes multiply to –1

They have no relation

Parallel lines have equal slopes; hence, the slope of the midsegment equals that of the base.

Explanation

Parallel lines have equal slopes; hence, the slope of the midsegment equals that of the base.

13) In △ABC, the midpoints of AB and AC are M(3,0) and N(0,4). The length of MN = __.

Use distance formula: √((3−0)²+(0−4)²) = √(9+16) = √25 = 5.

Explanation

Use distance formula: √((3−0)²+(0−4)²) = √(9+16) = √25 = 5.

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14) In △ABC, if the midsegment MN = 7 cm, then the base QR =

7 cm

10.5 cm

14 cm

21 cm

The base is twice the midsegment: QR = 2 × 7 = 14 cm.

Explanation

The base is twice the midsegment: QR = 2 × 7 = 14 cm.

15) If the midsegment theorem is applied in quadrilateral proofs, it shows the inner shape is a:

Rectangle

Parallelogram

Kite

Trapezoid

Joining midpoints of a quadrilateral always forms a parallelogram—Varignon’s theorem.

Explanation

Joining midpoints of a quadrilateral always forms a parallelogram—Varignon’s theorem.

16) In △PQR, if PQ = 12 cm, PR = 10 cm, and M, N are midpoints, then MN ∥ QR and MN = __ cm.

The midsegment parallel to QR equals half of QR, regardless of PQ and PR lengths.

Explanation

The midsegment parallel to QR equals half of QR, regardless of PQ and PR lengths.

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17) Select all figures where the midsegment rule applies directly.

Triangles

Quadrilaterals

Parallelograms

Trapezoids

Squares

Midsegments apply to triangles and quadrilaterals (Varignon’s theorem), forming similar triangles or parallelograms.

Explanation

Midsegments apply to triangles and quadrilaterals (Varignon’s theorem), forming similar triangles or parallelograms.

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18) The ratio of perimeters of the smaller triangle formed by midsegments to the original triangle is:

1:1

1:2

1:3

1:4

Each side is half as long, so the perimeter ratio is 1:2.

Explanation

Each side is half as long, so the perimeter ratio is 1:2.

19) The triangle formed by connecting midpoints of all sides is congruent to the original triangle.

True

False

It is similar, not congruent, because all sides are half as long but angles remain equal.

Explanation

It is similar, not congruent, because all sides are half as long but angles remain equal.

20) In △DEF, if DE = 5, EF = 8, and DF = 6, then the midsegment parallel to EF measures:

3

4

5

8

Half of EF = ½ × 8 = 4 units.

Explanation

Half of EF = ½ × 8 = 4 units.