Using similar triangles if KLM~ PQR with a scale factor of 3:5, find the perimeter of PQR (With work please)

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nicole723 nicole723

15-09-2018
Mathematics

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Using similar triangles if KLM~ PQR with a scale factor of 3:5, find the perimeter of PQR (With work please)

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JeanaShupp JeanaShupp · Answer 1 · 2019-01-18T10:02:38+01:00

Answer: 55 units

Step-by-step explanation:

Given: ΔKLM~ ΔPQR with a scale factor of 3:5

⇒[tex]\frac{sides\ of\ \triangle{KLM}}{sides\ of\ \triangle{PQR}}=\frac{3}{5}[/tex]

Also ΔKLM~ ΔPQR and the corresponding sides of similar triangles are proportional therefore,

[tex]\\\Rightarrow\frac{KL}{PQ}=\frac{LM}{QR}=\frac{KM}{PR}=\frac{3}{5}[/tex]

[tex]\\\Rightarrow PQ=\frac{KL\times5}{3}=\frac{6\times5}{3}=10[/tex]

Similarly,

[tex]QR=\frac{LM\times5}{3}=\frac{15\times5}{3}=25\\PR=\frac{KM\times5}{3}=\frac{12\times5}{3}=20[/tex]

Now, Perimeter of ΔPQR=[tex]PQ+QR+PR=25+20+10=55\ units[/tex]

MrRoyal MrRoyal · Answer 2 · 2021-10-25T15:57:48+02:00

Triangles KLM and PQR are similar triangles, but they are not congruent

The perimeter of triangle PQR is 55 units.

From the question, we have:

[tex]\mathbf{KL = 6}[/tex]

[tex]\mathbf{KM = 12}\\[/tex]

[tex]\mathbf{LM = 15}[/tex]

Because the scale ratio is given as 3:5.

So, we have:

[tex]\mathbf{KL: PQ = 3:5}[/tex]

[tex]\mathbf{KM:OP = 3 : 5}[/tex]

[tex]\mathbf{LM : QR = 3 : 5}[/tex]

Substitute 6 for KL in [tex]\mathbf{KL: PQ = 3:5}[/tex]

[tex]\mathbf{6: PQ = 3:5}[/tex]

Express as fractions

[tex]\mathbf{\frac{PQ}{6} = \frac{5}{3}}[/tex]

Multiply both sides by 6

[tex]\mathbf{PQ = \frac{5}{3} \times 6}[/tex]

[tex]\mathbf{PQ = 5 \times 2}[/tex]

[tex]\mathbf{PQ = 10}[/tex]

Substitute 12 for KM in [tex]\mathbf{KM:OP = 3 : 5}[/tex]

[tex]\mathbf{12:OP = 3 : 5}[/tex]

Express as fractions

[tex]\mathbf{\frac{OP}{12} = \frac{5}{3} }[/tex]

Multiply both sides by 12

[tex]\mathbf{OP = \frac{5}{3} \times 12}[/tex]

[tex]\mathbf{OP = 5 \times 4}[/tex]

[tex]\mathbf{OP = 20}[/tex]

Substitute 15 for LM in [tex]\mathbf{LM : QR = 3 : 5}[/tex]

[tex]\mathbf{15 : QR = 3 : 5}[/tex]

Express as fractions

[tex]\mathbf{\frac{QR}{15} = \frac{5}{3}}[/tex]

Multiply both sides by 15

[tex]\mathbf{QR = \frac{5}{3} \times 15}[/tex]

[tex]\mathbf{QR = 5 \times 5}[/tex]

[tex]\mathbf{QR = 25}[/tex]

So, we have:

[tex]\mathbf{PQ = 10}[/tex]

[tex]\mathbf{OP = 20}[/tex]

[tex]\mathbf{QR = 25}[/tex]

The perimeter of PQR is then calculated as:

[tex]\mathbf{Perimeter = PQ + OP + QR}[/tex]

[tex]\mathbf{Perimeter = 10 + 20 + 25}[/tex]

[tex]\mathbf{Perimeter = 55}[/tex]

Hence, the perimeter of triangle PQR is 55 units.