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Theorem 6.7 : If a perpendicular is draw...

Theorem 6.7 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Text Solution

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Proof `:` In `Delta PSQ` and `Delta PQR `
`/_QPS ~= /_ RPQ ` ...(Common angle )
`/_ PSQ ~= /_PQR ` …(Each of measure ` 90^(@)` )
`:. Delta PSQ ~= Delta PQR ` ....( AA test of similarity ) …(1)
In `Delta QSR ` and `Delta PQR `
`/_ SRQ ~= /_ QRP ` ....( Common angle )
`/_ QSR ~= /_ PQR ` ....( Each of meausre `90^(@)`)
`:. Delta QSR ~ Delta PQ ` ....(AA test of similarity ) ....(2)
From (1) and (2) , we get
`Delta PSQ ~ Delta QSR`
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Knowledge Check

  • In a right angled triangle, the hypotenuse is 2sqrt(2) times the length of the perpendicular drawn from the opposite vertex on its hypotenuse then the other two angles are

    A
    `pi//3, pi//6`
    B
    `pi//4, pi//4`
    C
    `pi//8, 3pi//8`
    D
    `pi//12, 5pi//12`

  • In a right angled triangle, if the hypotenuse is 2sqrt2 times the length of perpendicular drawn from the opposite vertex on the hypotenuse, then the other two angles are

    A
    `pi/3, pi/6`
    B
    `pi/4, pi/4`
    C
    `pi/8, (3pi)/8`
    D
    `pi/12, (5 pi)/12`