jugoslav

MR = NR  

PR * LR = MR * NR

40 * 16 = MR * NR

MN = 16√10

MR = NR  

PR * LR = MR * NR

40 * 16 = MR * NR

MN = 16√10

The golden rectangle is inscribed in triangle ABC so that the longer side of a rectangle lies on the side AC. If AB = 7, BC = 11, and AC = 14, then what's the area of a rectangle?

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Using the law of cosines, we find that ∠A ≈ 50.754º and ∠C ≈ 29.526º.

To calculate an x, we can use this equation:

(x / tanA) + (x * Φ) + (x / tanC) = 14                 Φ = 1/2(1 + √5)

x ≈ 3.333

Rectangle area = x²Φ ≈ 17.974

Oops! I didn't mean that...

I meant that ΔABE ~ ΔCDF

Square BCFE is inscribed in the right triangle AGD, as shown below. If AB = 28 units and CD = 68 units, what is the area of square BCFE?

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∠ABE ~ ∠CDF

x / 28 = 68 / x

x² = 1904

[BCFE] = 1904 u²

The circle centered at (2, -1) and with radius 4 intersects the circle centered at (2, 5) and with radius √10 at two points A and B. Find (AB)².

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Let the centers of the circles be   P(2, -1) and Q(2, 5)       PQ = |-1 + 5| = 6

Let the midpoint of AB be M.     AM = BM ==> x

sqrt(AP² - x²) + sqrt(AQ² - x²) = PQ              x = (√15) / 2

AB² = (2x)² 

AB² = 15

CC₂ = sqrt(40² + 50²)

CC₂ = 2√1025

Side = sqrt[(840 / 40)² + (40 / 2)²]

Side = 29

Line segment DX is the height of triangle DBC.    

[DBC] = 1/2(144)

DX = (2 * [DBC]) / BC

DX = 12

I think both of you are correct.  (At least regarding the word common  )

In Alan's case, the tangent is called a Common external tangent.    (or common direct tangent)

In your case, it's called a Common internal tangent.       (or common indirect tangent)

But if circles are tangent to each other, then there are 2 external and 1 internal tangent.

Which one of these two is more common? 

Btw, the first answer is mine.