Question

what is the value of \\(m\\) in the figure below? in this diagram, \\(\delta abd \sim \delta bcd\\).

a. \\(\sqrt{136}\\)
b. \\(\sqrt{274}\\)
c. \\(\sqrt{32}\\)
d. \\(\sqrt{632}\\)
e. \\(\sqrt{72}\\)
f. \\(\sqrt{147}\\)

Explanation:

Identify the given values and similarity relation

Using the Means of a Proportion and Extremes of a Proportion knowledge points
We are given:

• \(\triangle ABD \sim \triangle BCD\)
• \(AD = 9\)
• \(CD = 8\)
• \(BC = m\)
• \(\angle BDC = 90^\circ\) (indicated by the right-angle symbol at \(D\))

Set up the ratio from similarity

Using the Means of a Proportion and Extremes of a Proportion knowledge points
Since \(\triangle ABD \sim \triangle BCD\), the ratios of corresponding sides are equal:

$$ \frac{BD}{CD} = \frac{AD}{BD} $$

Substituting the known values:

$$ \frac{BD}{8} = \frac{9}{BD} $$

$$ BD^2 = 9 \cdot 8 = 72 $$

Apply the Pythagorean theorem to find m

Using the Pythagorean Theorem knowledge point
In the right triangle \(\triangle BCD\), the hypotenuse is \(BC = m\):

$$ m^2 = BD^2 + CD^2 $$

Substituting \(BD^2 = 72\) and \(CD = 8\):

$$ m^2 = 72 + 8^2 $$

$$ m^2 = 72 + 64 = 136 $$

$$ m = \sqrt{136} $$

Answer:

• A. \(\sqrt{136}\) (Correct answer)
• B. \(\sqrt{274}\)
• C. \(\sqrt{32}\)
• D. \(\sqrt{632}\)
• E. \(\sqrt{72}\)
• F. \(\sqrt{147}\)