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Triangles: Competency Practice

Basic proportionality applications, similarity criteria reasoning, and perimeter relations.

8 problems·20–25 min·★★★★☆
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  1. Problem 1
    Basic Proportionality Theorem·★★★★☆
    In triangle ABC, D lies on AB and E lies on AC with $DE\parallel BC$. Prove that $AD/DB=AE/EC$.
    Triangle with an internal segment parallel to its base
    ABCDENot to scale
    ▶Answer
    $AD/DB=AE/EC$.
    ▶Step-by-step solution
    1. Triangles ADE and BDE have the same altitude to AB, so their area ratio is $AD/DB$.
    2. Triangles ADE and CDE have the same altitude to AC, so their area ratio is $AE/EC$.
    3. Triangles BDE and CDE lie between the same parallels DE and BC and have equal areas; equating the two ratios gives $AD/DB=AE/EC$.
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  2. Problem 2
    Basic Proportionality Theorem·★★★★☆
    In triangle CAB, DE is parallel to AB. If $AD=x+18$, $DC=6$, $BE=16$ and $EC=4$, find $x$.
    ▶Answer
    $x=6$
    ▶Step-by-step solution
    1. A parallel segment divides the two sides proportionally.
    2. Thus $(x+18)/6=16/4$.
    3. Solving the linear equation gives $x=6$.
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  3. Problem 3
    Similar Triangles·★★★★☆
    D lies on AB in triangle ABC and $\angle ACB=\angle CDA$. If $AC=16$ cm and $AD=4$ cm, find DB.
    Triangle divided by a segment from a vertex to its base
    4?16ADBCNot to scale
    ▶Answer
    $DB=60\text{ cm}$
    ▶Step-by-step solution
    1. $\angle CAB=\angle CAD$ and $\angle ACB=\angle CDA$, so $\triangle ACB\sim\triangle ADC$.
    2. Hence $AC/AD=AB/AC$, so $AB=16^2/4=64$ cm.
    3. Therefore $DB=AB-AD=64-4=60$ cm.
    Triangle divided by a segment from a vertex to its base
    46016ADBCNot to scale
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  4. Problem 4
    Similar Triangles·★★★☆☆
    Triangles ABC and DEF are similar. If $DE=18$, $EF=24$, $FD=30$ and $AB=9$ cm, find the perimeter of ABC.
    Two triangles in corresponding order
    9??182430ABCDEFNot to scale
    ▶Answer
    $36\text{ cm}$
    ▶Step-by-step solution
    1. The scale factor from ABC to DEF is $DE/AB=18/9=2$.
    2. Therefore $BC=24/2=12$ and $CA=30/2=15$.
    3. The perimeter is $9+12+15=36$ cm.
    Two triangles in corresponding order
    91215182430ABCDEFNot to scale
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  5. Problem 5
    Similarity Criteria·★★★★☆
    One angle of a triangle equals one angle of another triangle, and two side pairs are proportional. Must the triangles be similar? State the missing condition.
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    Not always. The equal angle must be included between the proportional side pairs for SAS similarity.
    ▶Step-by-step solution
    1. SAS similarity uses two proportional side pairs and the angle between those sides.
    2. If the stated equal angle is not included, the data can describe different triangle shapes.
    3. Therefore similarity follows only when the equal angles are the included corresponding angles.
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  6. Problem 6
    Similarity Criteria·★★★★☆
    Two right triangles each have an acute angle of $28^\circ$. Must they be similar? Give a reason.
    ▶Answer
    Yes, they are similar by AA.
    ▶Step-by-step solution
    1. Both triangles contain a $90^\circ$ angle.
    2. They also contain matching $28^\circ$ angles.
    3. Two corresponding angles are equal, so AA similarity applies.
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  7. Problem 7
    Similarity Criteria·★★★★☆
    Two corresponding sides and the perimeter of triangle DEF are each 5 times those of triangle ABC. Must the triangles be similar? Explain.
    Two triangles in corresponding order
    512?2560?ABCDEFNot to scale
    ▶Answer
    Yes, the triangles are similar by SSS.
    ▶Step-by-step solution
    1. The two known side pairs have the same scale factor 5.
    2. Subtract those two sides from each perimeter; the remaining side pair has the same scale factor.
    3. All three corresponding sides are proportional, so SSS similarity applies.
    Two triangles in corresponding order
    51213256065ABCDEFNot to scale
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  8. Problem 8
    Basic Proportionality Theorem·★★★★☆
    Points D and E divide sides AB and AC of triangle ABC with $AD=4$, $DB=12$, $AE=3$ and $EC=10$. Is DE parallel to BC? Explain.
    Triangle with an internal segment parallel to its base
    412310ABCDENot to scale
    ▶Answer
    No.
    ▶Step-by-step solution
    1. Compute $AD/DB=4/12$.
    2. Compute $AE/EC=3/10$.
    3. The ratios are unequal, so the converse condition is not satisfied.
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