To find: The values of KP, KM, MR, ML, MN and PR. Given: PR―∣∣KL―=9,ln=16,PM=2(KP) KM=KP+PM=3KP

Conclusion: By A-A similarity, it can be proved that: △LMK∼△MNK Similar triangles have corresponding sides as proportional. MKLK=NKMK ⇒3KP25=93KP ⇒KP=5 ⇒KM=3KP=15 By A-A similarity, it can be proved that: △LMK∼△RMP Similar triangles have corresponding sides as proportional. MKLK=PMRP ⇒1525=10RP ⇒KP=1623 Using Pythagorean Theorem in right angled triangle △LMK: Hypotenuse2=Base2+Perpendicular2 =252=152+LM2 ⇒LM2=400 ⇒LM=20 In similar triangles: △LMK∼△MNK Ratio can be written as: LMMK=MNNK ⇒2015=MN9 ⇒MN=12 KP=5,KM=15,MR=1313,ML=20,MN=12,PR=1623

Expert answers to "To find: The values of KP, KM, MR,..."

Chaya Galloway

Answered question

2021-01-10

To find: The values of KP, KM, MR, ML, MN and PR.
Given:

$\overset{―}{P R} ∣ ∣ \overset{―}{K L} = 9, \ln = 16, P M = 2 (K P)$
KM=KP+PM=3KP

Answer & Explanation

funblogC

Skilled2021-01-11Added 91 answers

Conclusion:
By A-A similarity, it can be proved that:
$△ L M K \sim △ M N K$
Similar triangles have corresponding sides as proportional.
$\frac{M K}{L K} = \frac{N K}{M K}$
$\Rightarrow \frac{3 K P}{25} = \frac{9}{3 K P}$
$\Rightarrow K P = 5$
$\Rightarrow K M = 3 K P = 15$
By A-A similarity, it can be proved that:
$△ L M K \sim △ R M P$
Similar triangles have corresponding sides as proportional.
$\frac{M K}{L K} = \frac{P M}{R P}$
$\Rightarrow \frac{15}{25} = \frac{10}{R P}$
$\Rightarrow K P = 16 \frac{2}{3}$
Using Pythagorean Theorem in right angled triangle $△ L M K$ :
$H y p o t e n u s e^{2} = B a s e^{2} + P e r p e n d i c u l a r^{2}$
$= 25^{2} = 15^{2} + L M^{2}$
$\Rightarrow L M^{2} = 400$
$\Rightarrow L M = 20$
In similar triangles: $△ L M K \sim △ M N K$
Ratio can be written as:
$\frac{L M}{M K} = \frac{M N}{N K}$
$\Rightarrow \frac{20}{15} = \frac{M N}{9}$
$\Rightarrow M N = 12$
$K P = 5, K M = 15, M R = 13 \frac{1}{3}, M L = 20, M N = 12, P R = 16 \frac{2}{3}$

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