What is the value of x?

Question

1.] Given: Triangle ABC is similar to triangle DEF. Segment AR is a median in triangle ABC and segment DS is a median in triangle DEF.
AB = 2x + 5, DE = x + 7, AR = 24, DS = 18. What is the value of x?
2.] Given: RW is the angle bisector of triangle QRS.
QR = 6, QW = x, QS = 20, RS = 12, find x.
3.] Given:Triangle A B C is similar to triangle X Y Z~Triangle A B C is similar to triangle X Y Zwith altitudes CR and ZS.
ZS = 12, XY = 15, YZ = 18 and AB = 20, find the measure of CR.

Answer 1

1.) In similar triangles, all corresponding linear pieces are in the same ratio. So AR/DS = (2x + 5) / (x + 7). Substituting 24 for AR and 18 for DS, we obtain this equation:

24/18 = (2x + 5) / (x + 7). Cross-multiplying gets us:

24x + 168 = 36x + 90. Now we simply solve for x:

24x + 168 = 36x + 90
78 = 12x
78/12 = x
39/6 = x
13/2 = x

Try subbing this back into the original equation. It works.

2.) An angle bisector divides the side opposite the angle it bisects in the same ratio as the other two sides of the triangle. Furthermore, the relative lengths thus created will fall on the same side as the shorter or longer side. That is, the shorter length of this divided line will fall on the side of the triangle containing the shorter side and the longer length will fall on the side containing the longer side of the triangle. Given these facts and the information here, we can come up with this:

WS = QS - QW = 20 - x.

From this we can come up with this relationship:

(20 - x) / x = 12 / 6 = 2
(20 - x) / x = 2.

Now we can solve for x:

20 - x = 2x
20 = 3x
20/3 = x and 20 - x = 40/3

Sub either back into (20 - x) / x = 2 and they work.

3.) Again we simply use the fact that corresponding linear pieces of similar triangles are in the same ratio. Since the bases of these two triangles are in the ratio 20/15, then the altitudes are in the same ratio. We can then set up this equation:

20/15 = CR/12.

Cross-multiplying, we get:

240 = 15CR
240/15 = CR
16 = CR.

Answer 2

! shows factorial, the place you multiply a variety with the aid of each integer decrease than it. a million! = a million 2! = a million • 2 3! = a million • 2 • 3 for this reason, 10! = 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • a million = 3,628,800