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Find Length of CD with AB=21, BC=14, AD=54 (Collinear Points)
Mathematics
Grade 7 (Junior High)
Question Content
10. In the figure, it is given that AB = 21, BC = 14, and AD = 54. Find the length of CD. (Assume points A, B, C, D are collinear in order A–B–C–D.)
Correct Answer
19
Detailed Solution Steps
1
Step 1: Identify the relationship between the segments. If points A, B, C, D are collinear in the order A–B–C–D, then the total length AD is the sum of AB, BC, and CD. Mathematically, this is expressed as: \( AD = AB + BC + CD \).
2
Step 2: Rearrange the formula to solve for CD. Subtract AB and BC from both sides: \( CD = AD - AB - BC \).
3
Step 3: Substitute the given values (AB = 21, BC = 14, AD = 54) into the formula: \( CD = 54 - 21 - 14 \).
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Step 4: Calculate the result: \( 54 - 21 = 33 \); \( 33 - 14 = 19 \).
Knowledge Points Involved
1
Collinear Points and Segment Addition Postulate
If three or more points lie on a straight line (are collinear), the length of the entire segment is the sum of the lengths of its smaller segments. For collinear points A, B, C, D (in order), \( AD = AB + BC + CD \). This postulate helps relate the lengths of parts of a segment to the length of the whole.
2
Algebraic Manipulation for Solving Equations
To find an unknown length (e.g., \( CD \)), we rearrange the segment addition formula algebraically. Subtracting the known segment lengths (\( AB \), \( BC \)) from the total length (\( AD \)) isolates the unknown (\( CD \)), demonstrating how algebraic operations solve for unknowns in geometric contexts.
3
Arithmetic Operations (Subtraction)
Subtracting multiple values (21, 14) from a total (54) requires sequential or combined subtraction. This reinforces basic arithmetic skills applied to geometric problem-solving, where numerical values represent segment lengths.
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