Question

if a line is drawn parallel to one side of a triangle, it divides the other two sides:
a. diagonally
b. perpendicularly
c. equally
d. proportionally

a triangle has sides of length 5, 12, and 13. what type of triangle is this?
a. isosceles
b. obtuse
c. acute
d. right

in △abc, ∠acb = 90°, and cd is the altitude to ab. if ab = 20, ac = 12, and bc = 16, which of the following verifies the pythagorean theorem?
a. (12^2 + 16^2 = 19^2)
b. (12^2 + 16^2 = 20^2)
c. (12^2 + 16^2 = 22^2)
d. (12^2 + 16^2 = 18^2)

Explanation:

First Question:

The Basic Proportionality Theorem (Thales' theorem) states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. So the correct option is d.

To determine the triangle type, we check the Pythagorean theorem. For sides 5, 12, 13: \(5^2 + 12^2 = 25 + 144 = 169\) and \(13^2 = 169\). Since \(5^2 + 12^2 = 13^2\), it's a right triangle. So the correct option is d.

In a right triangle \( \triangle ABC \) with \( \angle ACB = 90^\circ \), the Pythagorean theorem is \( AC^2 + BC^2 = AB^2 \). Given \( AC = 12 \), \( BC = 16 \), \( AB = 20 \), we check \( 12^2 + 16^2 = 144 + 256 = 400 \) and \( 20^2 = 400 \). So \( 12^2 + 16^2 = 20^2 \), correct option is b.

Answer:

d. Proportionally

Second Question: