Two right triangles are graphed on a coordinate plane. one triangle has a vertical side of 4 and a horizontal side of 10. the other triangle has a vertical side of 12 and a horizontal side of 30.

could the hypotenuses of these two triangles lie along the same line?

no, because they are not similar triangles
no, because one is larger than the other
yes, because they are similar triangles
yes, because they are both right triangles

The answer is (c) Yes, because they are similar triangles.

Step-by-step explanation: We are given two right angled-triangles with vertical sides 4, 12 and horizontal sides 10, 30 units.

Let us consider ΔAOB and ΔPQR in the coordinate plane. Here, OA=4, OB=10, PQ=12 and QR=30.

Now,

Therefore, ΔOAB≈ΔPQR.

See the attached figure, if we draw the triangles with vertical sides parallel to  Y-axis and horizontal sides parallel to X-axis, then The hypotenuse AB and PR will lie on the same line.

Thus, the correct option is (c).

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4