Topic
For triangle ABC yes A(3 ; 7), REMOVE(–2 ; 2), OLD(sixty one). Write down the general equation of the altitudes of the triangle ABC.
Solution method – See details
Step 1: Find the VTPT coordinates of the altitudes that are the corresponding opposite sides
Step 2: Find the points that pass through the vertices of the triangle
Step 3: Write the PTTQ of the altitudes when the passing point and the corresponding VTPT are known
Detailed explanation
We have: \(\overrightarrow {AB} = ( – 5; – 5),\overrightarrow {AC} = (3; – 6),\overrightarrow {BC} = (8; – 1)\)
Call AH, BM, CN are the altitudes ofABC. Then:
+ \(AH \bot BC \Rightarrow \) AH pass A and get \(\overrightarrow {BC} = (8; – 1)\) as VTPT so PT: 8x – y – 17 = 0
+ \(BM \bot AC \Rightarrow \) BM pass REMOVE and get \(\overrightarrow {{n_1}} = (1; – 2)\) in the same direction as \(\overrightarrow {AC} = (3; – 6)\) as VTPT, so there is PT:
x – 2y + 6 = 0
+ \(CN \bot AB \Rightarrow \) CN pass OLD and get \(\overrightarrow {{n_2}} = (1;1)\) in the same direction as \(\overrightarrow {AB} = ( – 5; – 5)\) as VTPT, so there is PT:
x + y – 7 = 0
