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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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Explore conceptually related problems

If a perpendicular is drawn from the vertex containing the right angle of a right triangle to the hypotenuse then prove that the triangle on each side of the perpendicular are similar to each other and to the original triangle. Also, prove that the square of the perpendicular is equal to the product of the lengths of the two parts of the hypotenuse.

Show that in a right angled triangle, the hypotenuse is the longest side.

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`

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