Question

1. if triangles ade and abc shown in the figure below are similar, what is the value of x? a. 5 b. 10 c. 4 d. 8 11. find the value of x. a. 6 b. 10.7 c. 16 d. 22 12. in the figure the two triangles are similar. what is the value of x? a. 12.4 b. 22.1 c. 18.6 d. 13.2

Explanation:

Step1: Recall similar - triangle property

For similar triangles, the ratios of corresponding sides are equal.

Step2: Solve for \(x\) in the first problem (triangles \(ADE\) and \(ABC\))

Since \(\triangle ADE\sim\triangle ABC\), we have \(\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}\). Here, \(\frac{AD}{AB}=\frac{2}{2 + 4}=\frac{2}{6}=\frac{1}{3}\), and \(\frac{DE}{BC}=\frac{x}{12}\). Setting \(\frac{x}{12}=\frac{1}{3}\), we get \(x = 4\).

Step3: Solve for \(x\) in the second problem

The two right - angled triangles are similar. The ratio of the hypotenuses of the two similar right - angled triangles is \(\frac{24 - 8}{24}=\frac{16}{24}=\frac{2}{3}\). Let the shorter side of the smaller triangle be \(x\) and the shorter side of the larger triangle be \(18\). Then \(\frac{x}{18}=\frac{2}{3}\), so \(x=12\) (There seems to be an error in the options for this sub - question).

Step4: Solve for \(x\) in the third problem

Since the two triangles are similar, we have \(\frac{6}{11}=\frac{9}{19.8}=\frac{5}{x}\). Cross - multiplying \(\frac{5}{x}=\frac{6}{11}\), we get \(6x = 55\), then \(x=\frac{55}{6}\approx9.17\) (There seems to be an error in the options for this sub - question too).

Answer:

1. C. 4
2. (No correct option among given ones)
3. (No correct option among given ones)