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# Help!

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The line y = b-x with 0 < b < 4 intersects the y-axis at P and the line x= 4 at S. If the ratio of the area of triangle QRS to the area of triangle QOP is 9:25, what is the value of b? Express the answer as a decimal to the nearest tenth.

Oct 14, 2018

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Since the line  has the form  y  = -1x   + b....the slope is -1

This means OP  = OQ  and QR  = SR

If    [QRS ] / [QOP ]  = 9/25.....the scale factor is the square root of this =  3/5

This means that    QR : QO = 3 : 5

So...there are 8 equal parts to OR  and OQ  is 5 of them

So...the x coordinate of Q = the y coordinate of P  since OP  = OQ

And this coordinate is   ( 5 / [ 3 + 5 ] ) * 4 =  5/8 * 4  =  20/8  = 5/2 = 2.5

So...b = the y intercept P  = 5/2  =  2.5

Proof : [ QRS]  = (1/2)(QR) (SR)  = (1/2) (1.5)(1.5)  = (1/2)(2.25)

[QOP] =  (1/2) (OP)(OQ)  = (1/2)(2.5)(2.5)  = (1/2)(6.25)

[ QRS] / [ QOP ] =   2.25 / 6.25  =    9 / 25

See the graph, here :  https://www.desmos.com/calculator/m1ufgopozx

Oct 14, 2018