Mathematics: Given: △KLM, KM=48, LD=16, LD⊥KM, NOPS - rectangle NO:OP=5:9 Find: NO, OP

Given: △KLM, KM=48, LD=16, LD⊥KM, NOPS - rectangle

Ответ:

Alienhead6187

19.01.2022

Mathematics

$NO=10\ units\\ \\OP=18\ units$

Step-by-step explanation:

If NO:OP=5:9, then

$NO=5x\\ \\OP=9x$

Consider triangles LOP and LKM. In these triangles,

$\angle L\cong \angle L$ - reflexive property;

$\angle LOP\cong \angle LKM$ as corresponding angles when two parallel lines OP and KM are intersected by transversal KL;

$\angle LPO\cong \angle LMK$ as corresponding angles when two parallel lines OP and KM are intersected by transversal LM.

Hence, $\triangle LOP\sim \triangle LKM$ by AA similarity theorem. Similar triangles have proportional corresponding parts, so

$\dfrac{OP}{KM}=\dfrac{\text{Height }\triangle LOP}{\text{Height }\triangle LKM}\\ \\ \\\dfrac{9x}{48}=\dfrac{16-5x}{16}\\ \\9x\cdot 16=48\cdot (16-5x)\\ \\144x=768-240x\\ \\144x+240x=768\\ \\384x=768\\ \\x=2\\ \\NO=5x=5\cdot 2=10\ units\\ \\OP=9\cdot 2=18\ units$