Question

two triangles are similar, and one triangle has sides 8 cm, 12 cm, and 16 cm. the corresponding sides of the other triangle are proportional, with the shortest side measuring 6 cm. what is the length of the longest side of the second triangle?
a. 12 cm
b. 20 cm
c. 18 cm
d. 24 cm

a surveyor measures a triangular plot of land and finds two sides are 45 meters and 60 meters. the hypotenuse of the triangle is 75 meters. which theorem justifies this relationship?
a. sas similarity criterion
b. pythagorean theorem
c. aa similarity criterion
d. triangle proportionality theorem

Explanation:

Step1: Find the scale - factor

The shortest side of the first triangle is 8 cm and the shortest side of the second triangle is 6 cm. The scale - factor $k$ is $\frac{6}{8}=\frac{3}{4}$.

Step2: Find the longest side of the second triangle

The longest side of the first triangle is 16 cm. Let the longest side of the second triangle be $x$. Since the sides of similar triangles are proportional, we have $x = 16\times\frac{3}{4}$.
$x = 12$ cm

The Pythagorean theorem states that in a right - triangle, $a^{2}+b^{2}=c^{2}$, where $a$ and $b$ are the legs of the right - triangle and $c$ is the hypotenuse. Here, $45^{2}+60^{2}=2025 + 3600=5625$ and $75^{2}=5625$. So the Pythagorean theorem justifies the relationship.

Answer:

a. 12 cm