+1245 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.
+129852 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
