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Lines and Angles - Line Segment

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A line segment is a fixed portion of a line with two definite endpoints. Unlike a line, it has a measurable length. It is denoted as AB‾\overline{AB}.

A line segment AB with endpoints at A and B.
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Measuring a line segment is more accurate using a divider and a ruler than just a ruler. A divider helps avoid errors due to the thickness of the ruler or parallax error.

Diagram of a divider tool placed on a line segment for measurement.
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If a point CC lies between AA and BB, then the total length ABAB is the sum of the parts: AC+CB=ABAC + CB = AB.

Point C lying between A and B on a line segment.
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The Midpoint of a segment is a point that divides the segment into two equal halves. If MM is the midpoint of ABAB, then AM=MBAM = MB.

📐Formulae

AB+BC=ACAB + BC = AC (When BB lies between AA and CC)

AM=MB=12ABAM = MB = \frac{1}{2} AB (Where MM is the midpoint of AB‾\overline{AB})

💡Examples

Problem 1:

If points A,B,CA, B, C are on a line such that AB=5 cmAB = 5\text{ cm}, BC=3 cmBC = 3\text{ cm} and AC=8 cmAC = 8\text{ cm}, which point lies between the other two?

Solution:

We are given: AB=5 cmAB = 5\text{ cm} BC=3 cmBC = 3\text{ cm} AC=8 cmAC = 8\text{ cm} We observe that: 5 cm+3 cm=8 cm5\text{ cm} + 3\text{ cm} = 8\text{ cm} AB+BC=ACAB + BC = AC Therefore, point BB lies between AA and CC.

Explanation:

If the sum of two smaller segments equals the length of the longest segment, the common endpoint of the two smaller segments is the point that lies in the middle.

Problem 2:

Verify if DD is the midpoint of AG‾\overline{AG} on a number line where the coordinates of A,D,GA, D, G are 1,4,71, 4, 7 respectively.

Solution:

Length of AD=4−1=3AD = 4 - 1 = 3 units. Length of DG=7−4=3DG = 7 - 4 = 3 units. Since AD=DG=3 unitsAD = DG = 3\text{ units}, DD is the midpoint of AG‾\overline{AG}.

Explanation:

A midpoint divides a segment into two parts of equal length. Since the distance from AA to DD is equal to the distance from DD to GG, DD is the midpoint.

Problem 3:

A line segment XY‾\overline{XY} is 10 cm10\text{ cm} long. If a point ZZ is the midpoint of XY‾\overline{XY}, find the length of XZXZ.

Solution:

Total length XY=10 cmXY = 10\text{ cm}. Since ZZ is the midpoint, we use the formula: XZ=12XYXZ = \frac{1}{2} XY XZ=12×10XZ = \frac{1}{2} \times 10 XZ=5 cmXZ = 5\text{ cm}

Explanation:

The midpoint formula states that the distance from an endpoint to the midpoint is exactly half the total length of the segment.

Problem 4:

Given a line segment PQ‾\overline{PQ} of length 12 cm12\text{ cm}. If a point RR lies on PQ‾\overline{PQ} such that PR=4 cmPR = 4\text{ cm}, find the length of RQRQ.

Line segment PQ with point R between them.

Solution:

We know that if RR lies between PP and QQ, then: PR+RQ=PQPR + RQ = PQ Substituting the given values: 4+RQ=124 + RQ = 12 RQ=12−4RQ = 12 - 4 RQ=8 cmRQ = 8\text{ cm}

Explanation:

Since RR is on the segment PQPQ, the sum of the segments PRPR and RQRQ must equal the total length PQPQ. Subtracting the known part from the total gives the remaining part.

Problem 5:

In the following figure, if AD=9 cmAD = 9\text{ cm} and BB is the midpoint of ACAC, and CC is the midpoint of BDBD, find the lengths of AB,BC,AB, BC, and CDCD if they are all equal.

Line segment AD divided into three equal parts AB, BC, and CD.

Solution:

Let AB=BC=CD=xAB = BC = CD = x. Since A,B,C,DA, B, C, D are collinear in that order: AB+BC+CD=ADAB + BC + CD = AD x+x+x=9x + x + x = 9 3x=93x = 9 x=3x = 3 So, AB=3 cmAB = 3\text{ cm}, BC=3 cmBC = 3\text{ cm}, and CD=3 cmCD = 3\text{ cm}.

Explanation:

Because BB is the midpoint of ACAC, AB=BCAB = BC. Because CC is the midpoint of BDBD, BC=CDBC = CD. Therefore, AB=BC=CDAB = BC = CD. Since their sum is 99, each part is 99 divided by 33.