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In Each of the Figures Given Below, an Altitude is Drawn to the Hypotenuse by a Right-angled Triangle. the Length of Different Line-segment Are Marked in Each Figure. Determine X, Y, Z in Each Case. - Mathematics

Sum

In each of the figures given below, an altitude is drawn to the hypotenuse by a right-angled triangle. The length of different line-segment are marked in each figure. Determine x, y, z in each case.

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Solution

Δ PQR  is a right triangle, right angled at Q

`6+z^2 = (4+x)^2`

`6+z^2=16+x^2+8x`

`z^2-x^2-8x=16-36`

`z^2-x^2-8x=16-36`

`z^2-x^2-8x=-20`......(1)

Δ QSP  is a right triangle right angled at S

`QS^2+PS^2=PQ^2`

`y^2+4^2=6^2`

`y^2+16=36`

`y^2=36-16`

`y^2=20`

`y=sqrt20`

`y=sqrt(2xx2xx5)`

`y=2sqrt5`

Δ QSR  is a right triangle right angled at S

`QS^2+RS^2=QR^2`

`y^2+x^2=z^2`..........(2)

Now substituting  `y^2+x^2=z^2` in equation (i) we get

`y^2+x^2-x^2-8x=-20`

`20-8x=-20`

`-8x=-20-20`

`-8x=-40`

`x=40/8`

`x=5`

Now substituting ` x = 5` and  `y^2=20` in equation (ii) we get

`y^2+x^2=z^2`

`20+5^2=z^2`

`20+25=z^2`

`45=z^2`

`sqrt(3xx3xx5)=z^2`

`3sqrt5=z`

Hence the value of x, y and z are  `5,2sqrt5,3sqrt5`

Concept: Triangles Examples and Solutions
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APPEARS IN

RD Sharma Class 10 Maths
Chapter 7 Triangles
Q 21.2 | Page 126
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