Question

1. in a proportion if \\(\frac{a}{b} = \frac{c}{d}\\) then which of the following statements is not true?

a. \\(\frac{b}{a} = \frac{d}{c}\\) b. \\(\frac{a}{c} = \frac{b}{d}\\) c. \\(\frac{a + b}{b} = \frac{c + d}{d}\\) d. \\(\frac{a + d}{b} = \frac{b + c}{d}\\)

1. in the figure, there are three similar right triangles by right triangle proportionality theorem. which triangle is missing in this statement \\(hop \sim\\_\\_\\_\\_\sim oep\\)?

a. hoe
b. oeh
c. hop
d. heo

1. which statement is not true?

Explanation:

Question 3

To solve this, we use the Right Triangle Proportionality Theorem (also known as the Geometric Mean Theorem or Altitude-on-Hypotenuse Theorem). In a right triangle, when an altitude is drawn from the right angle to the hypotenuse, it creates two smaller similar triangles, and all three triangles (the original and the two smaller ones) are similar to each other.

Given the right triangle \( \triangle HOP \) with right angle at \( O \), and altitude \( OE \) (assuming \( E \) is on \( HP \)), we have:

• \( \triangle HOP \) (the original right triangle)
• \( \triangle OEP \) (one smaller right triangle)
• The missing triangle should be \( \triangle HEO \) (the other smaller right triangle) because \( \triangle HEO \sim \triangle HOP \sim \triangle OEP \) by AA similarity (all right triangles and share a common acute angle). Let's check the options:
• Option a: \( \triangle HOE \) – Not the standard notation.
• Option b: \( \triangle OEH \) – Same as \( \triangle HEO \) but ordered differently, but the notation in the problem is \( HOP \sim \_\_\_ \sim OEP \), so we need the triangle with vertices \( H, E, O \) in order to match similarity.
• Option c: \( \triangle HOP \) – This is the original, not the missing smaller one.
• Option d: \( \triangle HEO \) – Matches the similarity condition as it is a right triangle, shares \( \angle H \) with \( \triangle HOP \), and has a right angle at \( E \), so AA similarity holds.

Answer:

d. HEO