Question

which of the following best describes the triangle proportionality theorem?
a. if two triangles are similar, their corresponding angles are equal.
b. the sum of the interior angles of a triangle is 180 degrees.
c. if two triangles are congruent, their corresponding sides are proportional.
d. a line parallel to one side of a triangle divides the other two sides proportionally.

in triangle △abc, ab = 8 cm, bc = 10 cm, and ac = 12 cm. in triangle △def, de = 16 cm and ef = 20 cm. what is the length of df if △abc ~ △def?
a. 22 cm
b. 25 cm
c. 28 cm
d. 24 cm

the length of a buildings shadow is 20 meters at the same time that a 2 - meter stick casts a shadow of 4 meters. using similar triangles, what is the height of the building?
a. 5 meters
b. 15 meters
c. 10 meters
d. 20 meters

a tree casts a shadow of 12 feet, while a 4 - foot stick casts a shadow of 6 feet. what is the height of the tree?
a. 12 feet
b. 6 feet
c. 8 feet
d. 10 feet

Explanation:

First Question

The Triangle Proportionality Theorem (also known as Thales' theorem) states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides proportionally.

• Option a describes a property of similar triangles (angle - angle similarity).
• Option b is the angle - sum property of a triangle.
• Option c: Congruent triangles have equal sides (not just proportional).
• Option d matches the statement of the Triangle Proportionality Theorem.

Step1: Find the scale factor

Since \(\triangle ABC\sim\triangle DEF\), the scale factor \(k\) is given by \(\frac{DE}{AB}\). Given \(AB = 8\mathrm{cm}\) and \(DE=16\mathrm{cm}\), then \(k=\frac{DE}{AB}=\frac{16}{8} = 2\)

Step2: Use the scale factor to find \(DF\)

We know that \(AC\) and \(DF\) are corresponding sides. Given \(AC = 12\mathrm{cm}\), and using the formula for similar - triangle side lengths \(DF=k\times AC\)

Substitute \(k = 2\) and \(AC = 12\mathrm{cm}\) into the formula: \(DF=2\times12\)

Step1: Set up the proportion for similar triangles

Let \(h\) be the height of the building. For similar triangles, the ratio of height to shadow length is the same. So, \(\frac{h}{20}=\frac{2}{4}\)

Step2: Solve the proportion for \(h\)

Cross - multiply: \(4h=2\times20\)

Simplify the right - hand side: \(4h = 40\)

Divide both sides by \(4\): \(h=\frac{40}{4}\)

Answer:

d. A line parallel to one side of a triangle divides the other two sides proportionally.

Second Question