jugoslav

https://web2.0calc.com/questions/rhombus-abcd-has-perimeter-of-148-and-one-of-its

In a trapezoid ABCD with AB parallel to CD, the diagonals AC and BD intersect at E. If the area of triangle ABE is 50 units, and the area of triangle ADE is 40 square units, what is the area of trapezoid ABCD?

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ABCD quadrilateral is an isosceles trapezoid.

Let M and N be the midpoints on AB and CD.

Let AB = 20 and EM = 5

Angles EAM and EDN are congruent.

Let DN be "x"           tan( ∠EDN) = 0.5

So, we have              x(0.5 * x) = 40          x = 4√5        CD = 8√5

Height        MN = 5 + (0.5 * DN) = 9.472135955

[ABCD] = 1/2(AB + CD) * MN ≈ 179.443 u²

ABCD is a unit square.  E is the center of the square, and CF is tangent to the semicircle with diameter AD.  Find the area of triangle DEF.

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Let G be a midpoint on AD.

Quadrilateral CDGF is a kite.

Angles ADF and DCG are congruent.

DF = 2( sin∠DCG * CD) = 0.894427191

DE = 1/2(BD) = 0.707106781

∠EDF = 45º -  ∠ADF = 18.43494882º

Using the law of cosines, we can calculate the third side of the triangle DEF.

Using Heron's formula we can find the area of a triangle DEF.

[DEF] = 1/10 squared units

ABCD is a square with a side length of 1.  A circle passes through B, C, and is tangent to the midpoint of AD (at E).  What is the radius of the circle?

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AB = EF = 1      BF = 1/2         ∠BEF = tan^-1(1/2) = 26.56505118º

EB = sqrt(EF² + BF²) = √1.25

Radius      EO = 1/2 (EB / cos ∠BEF) = 0.625     or   5/8

A circle is centered at O  and has an area of 48pi. Let  Q and R  be points on the circle, and let P be the circumcenter of triangle QRO. If P is contained in triangle QRO  and triangle PQR  is equilateral, then find the area of triangle PQR.

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OQ = OR = r = 4√3

∠QOR = 30º         ∠QOP = 15º           ∠QPR = 60º

QR = 2 (OQ * sin∠QOP) ==>         QR = 3.586301889

Height of ΔPQR      h = sqrt[PQ² - (QR/2)²] = 3.105828541

[PQR] = h * QR / 2 = 5.569219382 u²

Circle A and Circle B intersect at two points P and Q. A common tangent line of the two circles touches circle A and circle B at points X and Y respectively. XY and the extension of PQ intersect at Z. We know the radius of circle A is 7, the radius of circle B is 24, and AB = 25. Find XZ / ZY.

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XZ ≅ ZY   ==>      XZ / ZY = 1  

In the figure, what is the area of triangle ABD? Express your answer as a common fraction.

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[ABD] = 5/13 [ABC]

Suppose that ABCD is a trapezoid in which AD is parallel to BC. Given AC is perpendicular to CD, AC bisects ∠BAD, and the area of ABCD is 20, then compute the area of triangle ACD.

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AB = BC = 1/2 (AD)

[ACD] = 2/3 (20) 

Sorry about that!  For some reason, I read ΔBCD as ABCD. 

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BC = CD = 5√6          ∠CBD = ∠BDC = 45º          ∠ADB = 105 - 45 = 60º

BD = 10√3            AD = cos(60º) * 10√3 = 5√3

AC = 16.73