Math, asked by cpvibhamenon, 8 months ago

P is the midpoint of hypothenuse AB in a right angled triangle ABC prove AB= 2CP find using midpoint theorems​

Answers

Answered by saounksh
3

ᴇxᴘʟᴀɪɴᴀᴛɪᴏɴ

ɢɪᴠᴇɴ

  • Right triangle ABC, right angled at C with P as the mid point of hypotenous AB.

ᴛᴏ ᴘʀᴏᴠᴇ

  • AB = 2CP

ᴄᴏɴsᴛʀᴜᴄᴛɪᴏɴ

  • Draw PD and PE parallel to BC and AC respectively.

ᴘʀᴏᴏғ

Mid Point Theorem

If a line is drawn from mid point of one side of a triangle, parallel to another side and intersects the third side then it will bisect the third side of the triangle.

By mid point theorem

AD = DC, BE = EC .....(1)

Also, PD||BC, PE||AC but AC⊥BC,

⇒PD ⊥ AC, PE ⊥BC

⇒ PDCE is a rectangle

⇒ DC = PE .....(2)

Again PE⊥BC, so ΔPCE is right triangle. Applying Pythagoras Theorem,

CP² = EC² + PE² .....(3)

Applying Pythagoras Theorem in ΔABC,

AB² = AC² + BC²

⇒AB² = (AD+DC)² + (BE+EC)²

⇒AB² = (2DC)² + (2EC)²[Using (1)]

⇒AB² = 4(DC)² + 4(EC)²

⇒AB² = 4(PE)² + 4(EC)²[Using (2)]

⇒AB² = 4[(PE)² + 4(EC)²]

⇒AB² = 4(CP)² [Using (3)]

⇒AB = 2CP

Hence Proved

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