Question

An infinite series of similar triangles to point c. If AE =16,Ed =8. What is the sum of all the vertical segment (AE+BD+.........)

Answer 1

Given:

An infinite series of similar triangles to point c. If AE =16,Ed =8.

To find:

What is the sum of all the vertical segment (AE+BD+.........)

Solution:

From given figure it's clear that,

Since all right triangles are similar,

⇒ Δ ABE ~ Δ EDB

⇒ EB/AE = BE/DB

⇒ AB/AE = BD/EB

Let EB/AE = x, then EB = AEx and BD = EBx = 16x²

⇒ AE = 16, BD = 16x²,

the next vertical segment is 16x² × x² = 16x⁴ and so on.

Sum of all vertical segments is an infinite geometric series.

Also, EB² = ED² + BD²

⇒ (16x)² = 82 + (16x²)2

⇒ 4x² = 1 + 4x⁴

⇒ (2x²) - 2(2x²) + 1 = 0

⇒ (2x² - 1)² = 0

⇒ x = 1/√2

∴ AE + BD + ........

= 16 + 16x² + 16x⁴ + ....

= 16/(1 - x²)

= 16/(1 - 1/2)

= 32

∴ AE + BD + ........ = 32

Answer 2

Step-by-step explanation:

here you go hope you understood